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Properties

Label 2-26e2-676.347-c0-0-0
Degree $2$
Conductor $676$
Sign $0.791 + 0.610i$
Analytic cond. $0.337367$
Root an. cond. $0.580833$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.278 − 0.960i)2-s + (−0.845 − 0.534i)4-s + (0.787 + 1.14i)5-s + (−0.748 + 0.663i)8-s + (0.692 + 0.721i)9-s + (1.31 − 0.438i)10-s + (0.948 − 0.316i)13-s + (0.428 + 0.903i)16-s + (−1.60 − 0.535i)17-s + (0.885 − 0.464i)18-s + (−0.0557 − 1.38i)20-s + (−0.326 + 0.859i)25-s + (−0.0402 − 0.999i)26-s + (0.444 − 1.53i)29-s + (0.987 − 0.160i)32-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)2-s + (−0.845 − 0.534i)4-s + (0.787 + 1.14i)5-s + (−0.748 + 0.663i)8-s + (0.692 + 0.721i)9-s + (1.31 − 0.438i)10-s + (0.948 − 0.316i)13-s + (0.428 + 0.903i)16-s + (−1.60 − 0.535i)17-s + (0.885 − 0.464i)18-s + (−0.0557 − 1.38i)20-s + (−0.326 + 0.859i)25-s + (−0.0402 − 0.999i)26-s + (0.444 − 1.53i)29-s + (0.987 − 0.160i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.791 + 0.610i$
Analytic conductor: \(0.337367\)
Root analytic conductor: \(0.580833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 676,\ (\ :0),\ 0.791 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131893917\)
\(L(\frac12)\) \(\approx\) \(1.131893917\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.278 + 0.960i)T \)
13 \( 1 + (-0.948 + 0.316i)T \)
good3 \( 1 + (-0.692 - 0.721i)T^{2} \)
5 \( 1 + (-0.787 - 1.14i)T + (-0.354 + 0.935i)T^{2} \)
7 \( 1 + (-0.799 - 0.600i)T^{2} \)
11 \( 1 + (0.0402 + 0.999i)T^{2} \)
17 \( 1 + (1.60 + 0.535i)T + (0.799 + 0.600i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.444 + 1.53i)T + (-0.845 - 0.534i)T^{2} \)
31 \( 1 + (0.748 - 0.663i)T^{2} \)
37 \( 1 + (1.96 + 0.319i)T + (0.948 + 0.316i)T^{2} \)
41 \( 1 + (0.511 + 0.218i)T + (0.692 + 0.721i)T^{2} \)
43 \( 1 + (-0.948 + 0.316i)T^{2} \)
47 \( 1 + (-0.568 - 0.822i)T^{2} \)
53 \( 1 + (1.41 - 1.25i)T + (0.120 - 0.992i)T^{2} \)
59 \( 1 + (0.632 + 0.774i)T^{2} \)
61 \( 1 + (-0.368 + 1.80i)T + (-0.919 - 0.391i)T^{2} \)
67 \( 1 + (-0.428 + 0.903i)T^{2} \)
71 \( 1 + (-0.278 + 0.960i)T^{2} \)
73 \( 1 + (-0.970 - 0.239i)T + (0.885 + 0.464i)T^{2} \)
79 \( 1 + (-0.568 - 0.822i)T^{2} \)
83 \( 1 + (0.970 + 0.239i)T^{2} \)
89 \( 1 + (0.568 - 0.983i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.706 - 0.0570i)T + (0.987 + 0.160i)T^{2} \)
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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

−10.81424198440595743351464765054, −10.02297885568324859668991896713, −9.221867950438815913663602810488, −8.177114349274876189947845895842, −6.87614097220856170947645796043, −6.10892169319721718129374521281, −4.99307244503627617508301142934, −3.92857736953914023421947783444, −2.71088247536159905637376415221, −1.86856103531518868500180108293, 1.52652352497759982578068936445, 3.62087925516767649982291800530, 4.57076410016455899448725292634, 5.39023839894101881961288810475, 6.44613284086743785373513682215, 6.94918302849897440921555517787, 8.551351911019897743907546479925, 8.758361546807178607821238432891, 9.586713801610437649478421865097, 10.61351954924744515082518856116

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