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Properties

Label 2-952-952.781-c1-0-75
Degree $2$
Conductor $952$
Sign $0.407 + 0.913i$
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  + (0.910 − 1.08i)2-s + (−1.13 + 1.96i)3-s + (−0.342 − 1.97i)4-s + (−0.297 − 0.515i)5-s + (1.09 + 3.01i)6-s + (−1.98 + 1.75i)7-s + (−2.44 − 1.42i)8-s + (−1.07 − 1.86i)9-s + (−0.828 − 0.147i)10-s + (1.45 − 2.52i)11-s + (4.26 + 1.56i)12-s − 0.637i·13-s + (0.0949 + 3.74i)14-s + 1.35·15-s + (−3.76 + 1.34i)16-s + (3.95 − 1.16i)17-s + ⋯
L(s)  = 1  + (0.643 − 0.765i)2-s + (−0.655 + 1.13i)3-s + (−0.171 − 0.985i)4-s + (−0.133 − 0.230i)5-s + (0.446 + 1.23i)6-s + (−0.748 + 0.662i)7-s + (−0.864 − 0.503i)8-s + (−0.358 − 0.621i)9-s + (−0.262 − 0.0465i)10-s + (0.440 − 0.762i)11-s + (1.23 + 0.451i)12-s − 0.176i·13-s + (0.0253 + 0.999i)14-s + 0.348·15-s + (−0.941 + 0.337i)16-s + (0.959 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24618 - 0.808195i\)
\(L(\frac12)\) \(\approx\) \(1.24618 - 0.808195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.910 + 1.08i)T \)
7 \( 1 + (1.98 - 1.75i)T \)
17 \( 1 + (-3.95 + 1.16i)T \)
good3 \( 1 + (1.13 - 1.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.297 + 0.515i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.45 + 2.52i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.637iT - 13T^{2} \)
19 \( 1 + (-3.15 + 1.81i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.02 + 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + (-8.87 - 5.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.35 + 5.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.63iT - 41T^{2} \)
43 \( 1 + 3.51iT - 43T^{2} \)
47 \( 1 + (-4.34 - 7.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.12 - 4.68i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.3 - 5.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.9 + 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.09iT - 71T^{2} \)
73 \( 1 + (3.83 + 2.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.55 - 2.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.84iT - 83T^{2} \)
89 \( 1 + (8.18 + 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.56iT - 97T^{2} \)
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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

−10.23152376751568204584016087142, −9.174821389887489282418329526545, −8.831224041715380360242059067142, −7.05286269253983680297847278381, −5.90897868038893260605075972459, −5.41340491299762316975716549506, −4.55470858676995357373965943181, −3.55162472053418954796821375624, −2.76775958366993598374980001229, −0.72316308720596130492028622358, 1.23284666296799059870370097582, 3.04004248356966003692955744360, 3.98358228916034620798468826402, 5.22655083201034565189869395765, 6.06988863610816410392343355505, 6.91871080151229315890999460231, 7.23175948027310597282580489836, 8.009971809555037879606808771825, 9.328251374781894733569279682968, 10.11027091330182898726318329753

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