Properties

Label 4-2475e2-1.1-c1e2-0-24
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 3·8-s + 2·11-s − 5·13-s − 2·14-s + 16-s − 7·17-s + 6·19-s − 2·22-s − 9·23-s + 5·26-s + 2·28-s − 2·29-s − 3·31-s + 32-s + 7·34-s − 37-s − 6·38-s + 3·41-s − 3·43-s + 2·44-s + 9·46-s − 15·47-s − 11·49-s − 5·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 1.06·8-s + 0.603·11-s − 1.38·13-s − 0.534·14-s + 1/4·16-s − 1.69·17-s + 1.37·19-s − 0.426·22-s − 1.87·23-s + 0.980·26-s + 0.377·28-s − 0.371·29-s − 0.538·31-s + 0.176·32-s + 1.20·34-s − 0.164·37-s − 0.973·38-s + 0.468·41-s − 0.457·43-s + 0.301·44-s + 1.32·46-s − 2.18·47-s − 1.57·49-s − 0.693·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$C_4$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 96 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 19 T + 210 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_4$ \( 1 - 26 T + 330 T^{2} - 26 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 181 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−8.628643817948658194570071567041, −8.521474006888790969725667195784, −7.84969153747613827118586622428, −7.70110331352294270143682811351, −7.40797897017903521963391535348, −6.75810103023475061259622347960, −6.49698543446002896184771665688, −6.15268644491334724835131033809, −5.78679081711090809415128897765, −5.00913840568452111606782184511, −4.78453486102628049964348391551, −4.55767445917603769575339660848, −3.80228194479297070615110283786, −3.31466555826181734454393692320, −2.88452615319723069103592260008, −2.17567053527461890238233053421, −1.89271293181906110898218316504, −1.37006264010089288963729943768, 0, 0, 1.37006264010089288963729943768, 1.89271293181906110898218316504, 2.17567053527461890238233053421, 2.88452615319723069103592260008, 3.31466555826181734454393692320, 3.80228194479297070615110283786, 4.55767445917603769575339660848, 4.78453486102628049964348391551, 5.00913840568452111606782184511, 5.78679081711090809415128897765, 6.15268644491334724835131033809, 6.49698543446002896184771665688, 6.75810103023475061259622347960, 7.40797897017903521963391535348, 7.70110331352294270143682811351, 7.84969153747613827118586622428, 8.521474006888790969725667195784, 8.628643817948658194570071567041

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