Properties

Degree 2
Conductor 31
Sign $-0.629 + 0.776i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 − 1.35i)2-s + (−2.32 − 1.03i)3-s + (1.02 + 3.16i)4-s + (1.24 − 2.16i)5-s + (2.93 + 5.08i)6-s + (1.07 − 1.19i)7-s + (0.944 − 2.90i)8-s + (2.31 + 2.57i)9-s + (−5.26 + 2.34i)10-s + (−0.717 − 0.152i)11-s + (0.883 − 8.40i)12-s + (0.198 + 1.88i)13-s + (−3.61 + 0.768i)14-s + (−5.13 + 3.73i)15-s + (−0.326 + 0.237i)16-s + (4.28 − 0.910i)17-s + ⋯
L(s)  = 1  + (−1.32 − 0.959i)2-s + (−1.34 − 0.597i)3-s + (0.513 + 1.58i)4-s + (0.558 − 0.967i)5-s + (1.19 + 2.07i)6-s + (0.405 − 0.449i)7-s + (0.333 − 1.02i)8-s + (0.772 + 0.858i)9-s + (−1.66 + 0.741i)10-s + (−0.216 − 0.0459i)11-s + (0.255 − 2.42i)12-s + (0.0549 + 0.522i)13-s + (−0.966 + 0.205i)14-s + (−1.32 + 0.964i)15-s + (−0.0817 + 0.0593i)16-s + (1.03 − 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $-0.629 + 0.776i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ -0.629 + 0.776i)$
$L(1)$  $\approx$  $0.133511 - 0.280085i$
$L(\frac12)$  $\approx$  $0.133511 - 0.280085i$
$L(\frac{3}{2})$   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 \( 1 + (-4.81 - 2.79i)T \)
good2 \( 1 + (1.86 + 1.35i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.32 + 1.03i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (-1.24 + 2.16i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.07 + 1.19i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (0.717 + 0.152i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.198 - 1.88i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-4.28 + 0.910i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.484 + 4.61i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (2.19 - 6.77i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.104 - 0.0757i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (4.21 + 7.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.73 - 2.99i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (0.0240 - 0.229i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (-6.50 + 4.72i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.83 - 4.26i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-8.68 - 3.86i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + 7.84T + 61T^{2} \)
67 \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.27 + 2.53i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (2.63 + 0.559i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (4.42 - 0.941i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-2.43 + 1.08i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (0.681 + 2.09i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.79 - 11.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−17.20239434336908949523521721763, −16.25757409502916935580679206252, −13.56161595118880043355780066316, −12.24896106010159871238148308966, −11.49677316992833898796817174529, −10.31712306705212862634028142470, −9.019462257302901746338478001628, −7.40334349263818985824558502958, −5.36466432647098909702482918333, −1.25116026881693198379894299837, 5.56940224862133110592340098817, 6.50447684464949805039729358218, 8.186334906577435185773998852218, 10.14875364090992357531118950640, 10.40691314721274135541632023182, 12.02175626228520112710779076517, 14.46482013116267318378340955297, 15.46937920600878073113721667185, 16.56348914007766398031830838452, 17.31993971380190430090177729115

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