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Properties

Label 2-952-952.373-c1-0-72
Degree $2$
Conductor $952$
Sign $0.622 - 0.782i$
Analytic cond. $7.60175$
Root an. cond. $2.75712$
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  + (−1.37 − 0.311i)2-s + (1.23 + 2.14i)3-s + (1.80 + 0.859i)4-s + (0.751 − 1.30i)5-s + (−1.03 − 3.33i)6-s + (2.55 + 0.683i)7-s + (−2.22 − 1.74i)8-s + (−1.55 + 2.69i)9-s + (−1.44 + 1.56i)10-s + (1.94 + 3.36i)11-s + (0.391 + 4.92i)12-s − 6.06i·13-s + (−3.31 − 1.73i)14-s + 3.71·15-s + (2.52 + 3.10i)16-s + (−4.04 + 0.785i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.220i)2-s + (0.713 + 1.23i)3-s + (0.902 + 0.429i)4-s + (0.336 − 0.582i)5-s + (−0.423 − 1.36i)6-s + (0.966 + 0.258i)7-s + (−0.786 − 0.618i)8-s + (−0.517 + 0.897i)9-s + (−0.456 + 0.493i)10-s + (0.585 + 1.01i)11-s + (0.113 + 1.42i)12-s − 1.68i·13-s + (−0.885 − 0.464i)14-s + 0.959·15-s + (0.630 + 0.776i)16-s + (−0.981 + 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ 0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45792 + 0.702679i\)
\(L(\frac12)\) \(\approx\) \(1.45792 + 0.702679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.311i)T \)
7 \( 1 + (-2.55 - 0.683i)T \)
17 \( 1 + (4.04 - 0.785i)T \)
good3 \( 1 + (-1.23 - 2.14i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.751 + 1.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.06iT - 13T^{2} \)
19 \( 1 + (-1.76 - 1.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.71 - 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.640T + 29T^{2} \)
31 \( 1 + (2.61 - 1.50i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.52 + 7.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.20iT - 41T^{2} \)
43 \( 1 + 5.70iT - 43T^{2} \)
47 \( 1 + (4.60 - 7.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.03 - 1.75i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.139 - 0.0805i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.30 + 5.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.20 - 3.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.95iT - 71T^{2} \)
73 \( 1 + (7.57 - 4.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.11 + 0.642i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.13iT - 83T^{2} \)
89 \( 1 + (-2.43 + 4.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−9.976424097759839061216511847858, −9.161415947344153457549543647307, −8.903491191258144742974715406338, −7.948983841877930176658958885695, −7.18480516000665947215846636888, −5.63340963613399248971430554202, −4.81199827589294179649969382964, −3.72883088206447308921631272841, −2.63274474358406959792345333838, −1.40141486364166347746446141272, 1.14337666708429430942052609898, 2.03671719828283726876947194129, 2.95530969606275276618284047888, 4.66806808371906779962732950429, 6.27392142589327176870548616327, 6.74877510579488929727595075852, 7.34861803497204107229593724569, 8.410200343773562002486250785491, 8.754347686146294857782106748666, 9.625107392228627254648580890216

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