Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.524 + 0.851i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (−1.75 − 1.27i)3-s + (0.309 − 0.951i)4-s + 1.89·5-s − 2.17·6-s + (−0.5 + 1.53i)7-s + (−0.309 − 0.951i)8-s + (0.533 + 1.64i)9-s + (1.53 − 1.11i)10-s + (−1.87 + 5.77i)11-s + (−1.75 + 1.27i)12-s + (1.34 + 0.973i)13-s + (0.5 + 1.53i)14-s + (−3.34 − 2.42i)15-s + (−0.809 − 0.587i)16-s + (−0.586 − 1.80i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−1.01 − 0.737i)3-s + (0.154 − 0.475i)4-s + 0.849·5-s − 0.887·6-s + (−0.188 + 0.581i)7-s + (−0.109 − 0.336i)8-s + (0.177 + 0.547i)9-s + (0.485 − 0.353i)10-s + (−0.565 + 1.74i)11-s + (−0.507 + 0.368i)12-s + (0.371 + 0.270i)13-s + (0.133 + 0.411i)14-s + (−0.862 − 0.626i)15-s + (−0.202 − 0.146i)16-s + (−0.142 − 0.438i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.524 + 0.851i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.524 + 0.851i)$
$L(1)$  $\approx$  $0.832002 - 0.464397i$
$L(\frac12)$  $\approx$  $0.832002 - 0.464397i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-5.42 + 1.23i)T \)
good3 \( 1 + (1.75 + 1.27i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - 1.89T + 5T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (1.87 - 5.77i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.34 - 0.973i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.586 + 1.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.12 + 2.26i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.88 + 8.87i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (7.89 - 5.73i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 0.864T + 37T^{2} \)
41 \( 1 + (3.03 - 2.20i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-5.63 + 4.09i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.5 + 1.08i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.492 + 1.51i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.07 + 0.783i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 4.11T + 61T^{2} \)
67 \( 1 + 6.86T + 67T^{2} \)
71 \( 1 + (-0.694 - 2.13i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.02 + 6.24i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.14 - 3.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.35 + 4.61i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.50 + 10.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.43 + 10.5i)T + (-78.4 - 57.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.67323419111010919676132135878, −13.36571448128826776252478164044, −12.54727077615934972650803947259, −11.82522483892338652397545133979, −10.49461969313687702058298681353, −9.355835620582684596941224229260, −7.12059455256164925323394830305, −6.04590581038875975849091816613, −4.91977911745576872572309769370, −2.15201030841526748047471868594, 3.70919692856038466472340497286, 5.57054349070511346417697025475, 5.95473384180407717815223696556, 7.921064382129750058596932948997, 9.714711700742624901831902750435, 10.78751850474869252197155493694, 11.66708515481722582843268540681, 13.46206390414532723985162988789, 13.74881179102646429802483455981, 15.50801018949678269430865678419

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