Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.989 + 0.141i$
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 + (1.47 + 0.658i)3-s + (0.309 + 0.951i)4-s + (−0.204 + 0.354i)5-s + (−0.809 − 1.40i)6-s + (0.809 − 0.898i)7-s + (0.309 − 0.951i)8-s + (−0.255 − 0.283i)9-s + (0.373 − 0.166i)10-s + (−2.41 − 0.513i)11-s + (−0.169 + 1.60i)12-s + (0.199 + 1.90i)13-s + (−1.18 + 0.251i)14-s + (−0.535 + 0.388i)15-s + (−0.809 + 0.587i)16-s + (−7.88 + 1.67i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.853 + 0.379i)3-s + (0.154 + 0.475i)4-s + (−0.0914 + 0.158i)5-s + (−0.330 − 0.572i)6-s + (0.305 − 0.339i)7-s + (0.109 − 0.336i)8-s + (−0.0851 − 0.0946i)9-s + (0.118 − 0.0526i)10-s + (−0.727 − 0.154i)11-s + (−0.0488 + 0.464i)12-s + (0.0554 + 0.527i)13-s + (−0.316 + 0.0671i)14-s + (−0.138 + 0.100i)15-s + (−0.202 + 0.146i)16-s + (−1.91 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.989 + 0.141i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.989 + 0.141i)$
$L(1)$  $\approx$  $0.830159 - 0.0589325i$
$L(\frac12)$  $\approx$  $0.830159 - 0.0589325i$
$L(\frac{3}{2})$   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 \( 1 + (-3.20 + 4.55i)T \)
good3 \( 1 + (-1.47 - 0.658i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (0.204 - 0.354i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.809 + 0.898i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (2.41 + 0.513i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.199 - 1.90i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (7.88 - 1.67i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (0.0307 - 0.292i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-1.96 + 6.05i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.56 - 5.49i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (2.71 + 4.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.97 - 0.877i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (0.558 - 5.31i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (-1.19 + 0.870i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.93 - 2.15i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-8.67 - 3.86i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 - 1.44T + 61T^{2} \)
67 \( 1 + (-4.02 + 6.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.51 - 6.12i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (6.70 + 1.42i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (5.91 - 1.25i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (4.61 - 2.05i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (0.362 + 1.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.44 + 10.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−15.02956037472713523690971133076, −13.94783745449034808809765591308, −12.82136317172185215027629815080, −11.29695402719586580516414022905, −10.42098733351210056216413393093, −9.030939917872944705788015356464, −8.325876286238585468646523632518, −6.77844516328932417724933988781, −4.34384084681549357606701331316, −2.66872410555293996329814660535, 2.49107696846506670613093101132, 5.06947911473599281462363766351, 6.87828920332265777958628356260, 8.181660265960004501360635714602, 8.783571692607109864294840380016, 10.28038412352384215958050113122, 11.58176710167826499021910524376, 13.16608395864406390729456526646, 13.94786030281714440911893178128, 15.28603176218181803339794178732

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