Properties

Label 4-1815e2-1.1-c3e2-0-6
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $11467.9$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·3-s − 5·4-s + 10·5-s + 6·6-s + 7-s − 3·8-s + 27·9-s + 10·10-s − 30·12-s + 47·13-s + 14-s + 60·15-s − 29·16-s − 98·17-s + 27·18-s − 75·19-s − 50·20-s + 6·21-s − 57·23-s − 18·24-s + 75·25-s + 47·26-s + 108·27-s − 5·28-s − 127·29-s + 60·30-s + ⋯
L(s)  = 1  + 0.353·2-s + 1.15·3-s − 5/8·4-s + 0.894·5-s + 0.408·6-s + 0.0539·7-s − 0.132·8-s + 9-s + 0.316·10-s − 0.721·12-s + 1.00·13-s + 0.0190·14-s + 1.03·15-s − 0.453·16-s − 1.39·17-s + 0.353·18-s − 0.905·19-s − 0.559·20-s + 0.0623·21-s − 0.516·23-s − 0.153·24-s + 3/5·25-s + 0.354·26-s + 0.769·27-s − 0.0337·28-s − 0.813·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(11467.9\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3294225,\ (\ :3/2, 3/2),\ 1)\)

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$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 - p T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 p T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - T + 594 T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 47 T + 4936 T^{2} - 47 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 98 T + 12063 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 75 T + 14294 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 57 T + 15296 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 127 T + 52308 T^{2} + 127 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 183 T + 63434 T^{2} + 183 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 229 T + 50446 T^{2} + 229 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 18 T + 137554 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 186 T + 137774 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 615 T + 258896 T^{2} + 615 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 927 T + 512576 T^{2} + 927 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 380 T + 129354 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 509 T + 489940 T^{2} - 509 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1138 T + 694662 T^{2} + 1138 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 273 T + 574298 T^{2} + 273 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 440 T + 815938 T^{2} - 440 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 973 T + 1173960 T^{2} - 973 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 15 T + 864574 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1288 T + 1792530 T^{2} + 1288 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 76 T + 1799074 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.610531212247136331178139926527, −8.598589875853202426457129744490, −8.035122545384632581259806876668, −7.60242878441684910899767958349, −7.16812094674502770078198575038, −6.53891211914455939643634306485, −6.34489166767670248644275643056, −6.09918410394423305947878629641, −5.16260555057876345865275581251, −5.11954280602755234543018670329, −4.41224792007397389748013505921, −4.24025502974766166786762098241, −3.58148254422038250591354585898, −3.33119060220014748184995613980, −2.68098720031604988372088964112, −2.01314547814520529775825780244, −1.78958960317689192300296613911, −1.36718175397376415773211921976, 0, 0, 1.36718175397376415773211921976, 1.78958960317689192300296613911, 2.01314547814520529775825780244, 2.68098720031604988372088964112, 3.33119060220014748184995613980, 3.58148254422038250591354585898, 4.24025502974766166786762098241, 4.41224792007397389748013505921, 5.11954280602755234543018670329, 5.16260555057876345865275581251, 6.09918410394423305947878629641, 6.34489166767670248644275643056, 6.53891211914455939643634306485, 7.16812094674502770078198575038, 7.60242878441684910899767958349, 8.035122545384632581259806876668, 8.598589875853202426457129744490, 8.610531212247136331178139926527

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