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Properties

Label 2-31e2-31.14-c1-0-10
Degree $2$
Conductor $961$
Sign $0.00955 + 0.999i$
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  + (−0.309 + 0.951i)2-s + (−1.89 + 2.10i)3-s + (0.809 + 0.587i)4-s + (−1.41 − 2.44i)6-s + (−0.418 + 3.97i)7-s + (−2.42 + 1.76i)8-s + (−0.522 − 4.97i)9-s + (−2.58 + 1.15i)11-s + (−2.76 + 0.588i)12-s + (1.38 + 0.294i)13-s + (−3.65 − 1.62i)14-s + (−0.309 − 0.951i)16-s + (−1.29 − 0.575i)17-s + (4.89 + 1.03i)18-s + (3.91 − 0.831i)19-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.09 + 1.21i)3-s + (0.404 + 0.293i)4-s + (−0.577 − 0.999i)6-s + (−0.158 + 1.50i)7-s + (−0.858 + 0.623i)8-s + (−0.174 − 1.65i)9-s + (−0.779 + 0.346i)11-s + (−0.798 + 0.169i)12-s + (0.383 + 0.0815i)13-s + (−0.976 − 0.434i)14-s + (−0.0772 − 0.237i)16-s + (−0.313 − 0.139i)17-s + (1.15 + 0.245i)18-s + (0.897 − 0.190i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00955 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00955 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.00955 + 0.999i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.00955 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477567 - 0.473023i\)
\(L(\frac12)\) \(\approx\) \(0.477567 - 0.473023i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.89 - 2.10i)T + (-0.313 - 2.98i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.418 - 3.97i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (2.58 - 1.15i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-1.38 - 0.294i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (1.29 + 0.575i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-3.91 + 0.831i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (4.57 - 3.32i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.437 + 1.34i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (2.12 + 3.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.33 + 1.48i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (-8.29 + 1.76i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (-3.70 - 11.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.443 - 4.21i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-5.35 + 5.94i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.836 + 7.95i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (-3.87 + 1.72i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (-10.3 - 4.60i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-9.46 - 10.5i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (5.72 + 4.15i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.47 + 4.70i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.83877315101368497309490341188, −9.559032662429573092952990385870, −9.216434944139012975112134388248, −8.164000552601992227948529800664, −7.19237558695368592651088892243, −6.03045888342874098772954742485, −5.65328580767846694534502396537, −4.91149932881845190906234437916, −3.54181380540303562117851888183, −2.41959698738421559777251983737, 0.41615967259475292053876208177, 1.19967184586908385755097911037, 2.46754116402761816738830548634, 3.82523691270712967697827023142, 5.22820309411181465677200146852, 6.16592004785313063768460803564, 6.79105110897895366601007251829, 7.46988308013988823908073153063, 8.384925314968503849948991975864, 9.891377691549306531452186023308

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