Properties

Degree 2
Conductor 31
Sign $0.612 + 0.790i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.640 − 1.97i)2-s + (−1.43 + 1.58i)3-s + (−1.85 − 1.34i)4-s + (−1.17 + 2.03i)5-s + (2.21 + 3.83i)6-s + (0.384 − 3.65i)7-s + (−0.492 + 0.357i)8-s + (−0.164 − 1.56i)9-s + (3.25 + 3.61i)10-s + (−3.91 + 1.74i)11-s + (4.79 − 1.02i)12-s + (2.04 + 0.433i)13-s + (−6.95 − 3.09i)14-s + (−1.55 − 4.77i)15-s + (−1.02 − 3.16i)16-s + (1.94 + 0.865i)17-s + ⋯
L(s)  = 1  + (0.452 − 1.39i)2-s + (−0.826 + 0.917i)3-s + (−0.927 − 0.674i)4-s + (−0.525 + 0.909i)5-s + (0.904 + 1.56i)6-s + (0.145 − 1.38i)7-s + (−0.174 + 0.126i)8-s + (−0.0549 − 0.522i)9-s + (1.02 + 1.14i)10-s + (−1.17 + 0.524i)11-s + (1.38 − 0.294i)12-s + (0.566 + 0.120i)13-s + (−1.85 − 0.827i)14-s + (−0.400 − 1.23i)15-s + (−0.256 − 0.790i)16-s + (0.471 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.612 + 0.790i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (14, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.612 + 0.790i)$
$L(1)$  $\approx$  $0.609586 - 0.298700i$
$L(\frac12)$  $\approx$  $0.609586 - 0.298700i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 31$, \(F_p\) is a polynomial of degree 2. If $p = 31$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad31 \( 1 + (1.44 - 5.37i)T \)
good2 \( 1 + (-0.640 + 1.97i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.43 - 1.58i)T + (-0.313 - 2.98i)T^{2} \)
5 \( 1 + (1.17 - 2.03i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.384 + 3.65i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (3.91 - 1.74i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-2.04 - 0.433i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (-1.94 - 0.865i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-0.606 + 0.128i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-2.71 + 1.97i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.425 - 1.31i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (0.137 + 0.237i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.86 + 3.17i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (0.263 - 0.0560i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (-1.66 - 5.11i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.993 + 9.45i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (3.89 - 4.33i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 - 2.22T + 61T^{2} \)
67 \( 1 + (-6.80 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.139 - 1.32i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (12.9 - 5.76i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (-7.92 - 3.52i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (3.46 + 3.85i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (4.05 + 2.94i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.43 + 3.94i)T + (29.9 + 92.2i)T^{2} \)
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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

−16.76724443140689411085321905646, −15.64858361185838965900727881179, −14.16511633413436284441181428647, −12.88573465374776811806619311267, −11.35735009495122135241975711641, −10.65510858418630298559282187221, −10.17804945075337557201204846081, −7.32048335407830026030761745732, −4.82977166522661922656485005360, −3.53382860905873253360409113508, 5.22044426614243068574469174912, 5.98121444488268926644646859397, 7.65192480263642494626808025496, 8.663068288699736717001581085860, 11.42220323880679262954785029753, 12.54797205942435042824465670185, 13.40307010775868564518892363878, 15.16512797702226687018023386905, 15.94243066653705297095263669373, 16.87740997285511557477585924006

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