Properties

Label 2-416-1.1-c1-0-6
Degree $2$
Conductor $416$
Sign $1$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.81·3-s + 2.70·5-s − 3.36·7-s + 0.298·9-s + 5.17·11-s + 13-s + 4.90·15-s + 6.70·17-s − 5.17·19-s − 6.10·21-s + 2.29·25-s − 4.90·27-s − 2·29-s − 8.80·31-s + 9.40·33-s − 9.08·35-s + 2.70·37-s + 1.81·39-s + 3.40·41-s + 8.53·43-s + 0.806·45-s − 3.36·47-s + 4.29·49-s + 12.1·51-s − 11.4·53-s + 13.9·55-s − 9.40·57-s + ⋯
L(s)  = 1  + 1.04·3-s + 1.20·5-s − 1.27·7-s + 0.0994·9-s + 1.56·11-s + 0.277·13-s + 1.26·15-s + 1.62·17-s − 1.18·19-s − 1.33·21-s + 0.459·25-s − 0.944·27-s − 0.371·29-s − 1.58·31-s + 1.63·33-s − 1.53·35-s + 0.444·37-s + 0.290·39-s + 0.531·41-s + 1.30·43-s + 0.120·45-s − 0.490·47-s + 0.614·49-s + 1.70·51-s − 1.56·53-s + 1.88·55-s − 1.24·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $1$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.117534726\)
\(L(\frac12)\) \(\approx\) \(2.117534726\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - T \)
good3 \( 1 - 1.81T + 3T^{2} \)
5 \( 1 - 2.70T + 5T^{2} \)
7 \( 1 + 3.36T + 7T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
17 \( 1 - 6.70T + 17T^{2} \)
19 \( 1 + 5.17T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8.80T + 31T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 - 3.40T + 41T^{2} \)
43 \( 1 - 8.53T + 43T^{2} \)
47 \( 1 + 3.36T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 + 3.40T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 + 1.54T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.07346018747580726469529385500, −9.869216570009413572272856148872, −9.401271125912948868648489118613, −8.825347872487271081578967918482, −7.53855917356795547010575450045, −6.32477399448814862735134204577, −5.81769980246510077698773678534, −3.90867891403886856696370633029, −3.05473047718636732014897962769, −1.73478642937009357582373617156, 1.73478642937009357582373617156, 3.05473047718636732014897962769, 3.90867891403886856696370633029, 5.81769980246510077698773678534, 6.32477399448814862735134204577, 7.53855917356795547010575450045, 8.825347872487271081578967918482, 9.401271125912948868648489118613, 9.869216570009413572272856148872, 11.07346018747580726469529385500

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