Properties

Label 4-1575e2-1.1-c3e2-0-12
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $8635.61$
Root an. cond. $9.63991$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 5·4-s + 14·7-s − 33·8-s + 31·11-s − 39·13-s + 42·14-s − 21·16-s − 79·17-s − 56·19-s + 93·22-s + 254·23-s − 117·26-s − 70·28-s + 62·29-s − 135·31-s + 87·32-s − 237·34-s − 113·37-s − 168·38-s − 235·41-s − 804·43-s − 155·44-s + 762·46-s + 152·47-s + 147·49-s + 195·52-s + ⋯
L(s)  = 1  + 1.06·2-s − 5/8·4-s + 0.755·7-s − 1.45·8-s + 0.849·11-s − 0.832·13-s + 0.801·14-s − 0.328·16-s − 1.12·17-s − 0.676·19-s + 0.901·22-s + 2.30·23-s − 0.882·26-s − 0.472·28-s + 0.397·29-s − 0.782·31-s + 0.480·32-s − 1.19·34-s − 0.502·37-s − 0.717·38-s − 0.895·41-s − 2.85·43-s − 0.531·44-s + 2.44·46-s + 0.471·47-s + 3/7·49-s + 0.520·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8635.61\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good2$C_4$ \( 1 - 3 T + 7 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 31 T + 2796 T^{2} - 31 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 3 p T + 3546 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 79 T + 8288 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 56 T + 1174 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 254 T + 38015 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 62 T + 19751 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 135 T + 63420 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 113 T + 102964 T^{2} + 113 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 235 T + 143042 T^{2} + 235 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 804 T + 306321 T^{2} + 804 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 152 T + 180510 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 149 T + 269640 T^{2} - 149 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 441 T + 273734 T^{2} - 441 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 223 T + 149646 T^{2} + 223 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1157 T + 871890 T^{2} + 1157 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 619 T + 576906 T^{2} + 619 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 268 T + 763078 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 427 T + 986572 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1211 T + 720720 T^{2} - 1211 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 466 T + 306170 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 172 T - 273626 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.765312914968619364598855332762, −8.689257424532438980298814428305, −8.230175594864463793772414311778, −7.55548946289411703870080051325, −7.00040822047829722643853033199, −6.97305647109596391661098159589, −6.34234235654288878296059844512, −5.94215564109347423173408692149, −5.18093353796941387762300031023, −5.06437060952825914625600459184, −4.74839546286551991652220156230, −4.40788446758935621230015993616, −3.79294837469130521737204776784, −3.50723518173021526010334787445, −2.87729792334582457553884468573, −2.34758180111414412280306954723, −1.59212372667214328464560684939, −1.17605116690899426035567815458, 0, 0, 1.17605116690899426035567815458, 1.59212372667214328464560684939, 2.34758180111414412280306954723, 2.87729792334582457553884468573, 3.50723518173021526010334787445, 3.79294837469130521737204776784, 4.40788446758935621230015993616, 4.74839546286551991652220156230, 5.06437060952825914625600459184, 5.18093353796941387762300031023, 5.94215564109347423173408692149, 6.34234235654288878296059844512, 6.97305647109596391661098159589, 7.00040822047829722643853033199, 7.55548946289411703870080051325, 8.230175594864463793772414311778, 8.689257424532438980298814428305, 8.765312914968619364598855332762

Graph of the $Z$-function along the critical line