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Properties

Label 2-952-952.339-c1-0-92
Degree $2$
Conductor $952$
Sign $-0.996 - 0.0818i$
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  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−3.27 + 1.89i)5-s + (0.548 − 2.58i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (4.63 + 2.67i)10-s + (−3.55 + 1.15i)14-s + (−2.00 − 3.46i)16-s + (3.57 + 2.06i)17-s + (2.12 − 3.67i)18-s + (−5.69 + 3.28i)19-s − 7.56i·20-s + (−3.92 − 6.79i)23-s + (4.65 − 8.06i)25-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.46 + 0.845i)5-s + (0.207 − 0.978i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (1.46 + 0.845i)10-s + (−0.950 + 0.309i)14-s + (−0.500 − 0.866i)16-s + (0.866 + 0.499i)17-s + (0.499 − 0.866i)18-s + (−1.30 + 0.753i)19-s − 1.69i·20-s + (−0.817 − 1.41i)23-s + (0.931 − 1.61i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $-0.996 - 0.0818i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ -0.996 - 0.0818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00677387 + 0.165267i\)
\(L(\frac12)\) \(\approx\) \(0.00677387 + 0.165267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
7 \( 1 + (-0.548 + 2.58i)T \)
17 \( 1 + (-3.57 - 2.06i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.27 - 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
19 \( 1 + (5.69 - 3.28i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.92 + 6.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.97T + 29T^{2} \)
31 \( 1 + (5.43 + 3.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.74 + 4.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.45 + 1.41i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.1 - 7.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.10 + 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.65 - 2.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (-9.15 + 5.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.18670605761037222633051993580, −8.524731988710803160986596357152, −7.969684316349015353637077201496, −7.46137872653981381926267796679, −6.52200772299085642434155316609, −4.58863875502393952225106558580, −4.07637244453260583189739543659, −3.21409614452469009710127159072, −1.84147027597056521457203924839, −0.099337124313018937889377298346, 1.37908954730682503814435216010, 3.45162331566273569651096883737, 4.53798096351921195220062228705, 5.21213874523736940249976057772, 6.34553299972264070184166536043, 7.23094251243539856910268407141, 8.087507819808620198708173138502, 8.644660169149088764522311942177, 9.291454575394029461843351373985, 10.19345904041858290312304312478

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