Properties

Label 2-31e2-31.7-c1-0-34
Degree $2$
Conductor $961$
Sign $-0.153 - 0.988i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 1.53i)2-s + (0.978 − 0.207i)3-s + (−0.5 + 0.363i)4-s + (0.190 − 0.330i)5-s + (0.809 + 1.40i)6-s + (2.74 + 1.22i)7-s + (1.80 + 1.31i)8-s + (−1.82 + 0.813i)9-s + (0.604 + 0.128i)10-s + (0.547 + 5.20i)11-s + (−0.413 + 0.459i)12-s + (−1.24 − 1.37i)13-s + (−0.507 + 4.82i)14-s + (0.118 − 0.363i)15-s + (−1.50 + 4.61i)16-s + (0.442 − 4.21i)17-s + ⋯
L(s)  = 1  + (0.353 + 1.08i)2-s + (0.564 − 0.120i)3-s + (−0.250 + 0.181i)4-s + (0.0854 − 0.147i)5-s + (0.330 + 0.572i)6-s + (1.03 + 0.461i)7-s + (0.639 + 0.464i)8-s + (−0.609 + 0.271i)9-s + (0.191 + 0.0406i)10-s + (0.165 + 1.57i)11-s + (−0.119 + 0.132i)12-s + (−0.344 − 0.382i)13-s + (−0.135 + 1.29i)14-s + (0.0304 − 0.0937i)15-s + (−0.375 + 1.15i)16-s + (0.107 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.153 - 0.988i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77644 + 2.07337i\)
\(L(\frac12)\) \(\approx\) \(1.77644 + 2.07337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.978 + 0.207i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (-0.190 + 0.330i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.74 - 1.22i)T + (4.68 + 5.20i)T^{2} \)
11 \( 1 + (-0.547 - 5.20i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (1.24 + 1.37i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.442 + 4.21i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-3.34 + 3.71i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (2.80 + 2.04i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.97 + 6.06i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (-2.11 - 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.41 - 0.513i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (1.59 - 1.77i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (1.73 - 5.34i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.646 + 0.288i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (0.516 - 0.109i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (0.118 - 0.204i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.1 - 4.51i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (1.20 + 11.4i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (6.93 + 1.47i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 + (-6.97 + 5.06i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-15.1 + 10.9i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02210072621597689957686398344, −9.206286581503346255949415290352, −8.297292113873300619880767697851, −7.57731832057241107680884979617, −7.13441908420129117113620572991, −5.86595527908330031040092920227, −5.01937317403418002472148323647, −4.57506822597145748342164387213, −2.73136970837078241374868833766, −1.82131486956463051310435779230, 1.18893176681751076926950158809, 2.33158367423491637773942860434, 3.47383800564050540754393063253, 3.90991133505330854836096114206, 5.23514302420568479842552909160, 6.23603367592406476554416452818, 7.52009850035143660350602945880, 8.242323266400291895570192342123, 8.977344708401509629257355222115, 10.10178940174725705057293750257

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