Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.978 + 0.207i$
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 + (1.25 + 0.914i)3-s + (0.309 − 0.951i)4-s − 4.13·5-s + 1.55·6-s + (−0.5 + 1.53i)7-s + (−0.309 − 0.951i)8-s + (−0.179 − 0.551i)9-s + (−3.34 + 2.43i)10-s + (1.14 − 3.51i)11-s + (1.25 − 0.914i)12-s + (3.20 + 2.32i)13-s + (0.5 + 1.53i)14-s + (−5.20 − 3.78i)15-s + (−0.809 − 0.587i)16-s + (1.27 + 3.93i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.726 + 0.527i)3-s + (0.154 − 0.475i)4-s − 1.84·5-s + 0.635·6-s + (−0.188 + 0.581i)7-s + (−0.109 − 0.336i)8-s + (−0.0596 − 0.183i)9-s + (−1.05 + 0.768i)10-s + (0.343 − 1.05i)11-s + (0.363 − 0.263i)12-s + (0.888 + 0.645i)13-s + (0.133 + 0.411i)14-s + (−1.34 − 0.976i)15-s + (−0.202 − 0.146i)16-s + (0.309 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.978 + 0.207i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.978 + 0.207i)$
$L(1)$  $\approx$  $1.11658 - 0.117005i$
$L(\frac12)$  $\approx$  $1.11658 - 0.117005i$
$L(\frac{3}{2})$   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 \( 1 + (-2.85 + 4.78i)T \)
good3 \( 1 + (-1.25 - 0.914i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 4.13T + 5T^{2} \)
7 \( 1 + (0.5 - 1.53i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.14 + 3.51i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-3.20 - 2.32i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.27 - 3.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.62 - 2.63i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.306 + 0.944i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.00 - 2.18i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 + (-1.84 + 1.34i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (0.399 - 0.290i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.5 + 1.08i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.0524 + 0.161i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.1 + 7.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 3.34T + 61T^{2} \)
67 \( 1 - 1.47T + 67T^{2} \)
71 \( 1 + (-2.55 - 7.87i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.547 - 1.68i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.87 + 5.77i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.35 + 4.61i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (5.81 - 17.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.291 - 0.898i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.03545611675810170439482063700, −14.11403293300034102366923957140, −12.56648497941368419481259669619, −11.69418117705498165616334937112, −10.75737000977214053159066424054, −8.943367631792106227957380668089, −8.208331968190805234130787465840, −6.25572019484939935429637141249, −4.06637469604218327145211726045, −3.42614523253195833193741103486, 3.31361775191376484645838891897, 4.59140907646697592628821373139, 7.00759551180591823631475234829, 7.66871672961611368885983473463, 8.682675528728950075349764145989, 10.81500744471092254649233692798, 11.99274317553965789807614707208, 12.94574045651679090736445001780, 14.00041563904659805155379749685, 15.12654262686297396307749234109

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