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Properties

Label 2-952-7.4-c1-0-3
Degree $2$
Conductor $952$
Sign $-0.910 + 0.414i$
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  + (−1.18 + 2.04i)3-s + (0.949 + 1.64i)5-s + (−1.39 + 2.24i)7-s + (−1.29 − 2.24i)9-s + (−1.88 + 3.26i)11-s − 0.189·13-s − 4.49·15-s + (0.5 − 0.866i)17-s + (0.440 + 0.762i)19-s + (−2.94 − 5.52i)21-s + (1.98 + 3.43i)23-s + (0.695 − 1.20i)25-s − 0.961·27-s − 6.30·29-s + (2.94 − 5.09i)31-s + ⋯
L(s)  = 1  + (−0.682 + 1.18i)3-s + (0.424 + 0.735i)5-s + (−0.528 + 0.848i)7-s + (−0.432 − 0.748i)9-s + (−0.568 + 0.984i)11-s − 0.0526·13-s − 1.16·15-s + (0.121 − 0.210i)17-s + (0.101 + 0.174i)19-s + (−0.643 − 1.20i)21-s + (0.413 + 0.716i)23-s + (0.139 − 0.240i)25-s − 0.185·27-s − 1.16·29-s + (0.528 − 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $-0.910 + 0.414i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ -0.910 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178845 - 0.824986i\)
\(L(\frac12)\) \(\approx\) \(0.178845 - 0.824986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.39 - 2.24i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (1.18 - 2.04i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.949 - 1.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.88 - 3.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.189T + 13T^{2} \)
19 \( 1 + (-0.440 - 0.762i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.98 - 3.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.30T + 29T^{2} \)
31 \( 1 + (-2.94 + 5.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.44 - 9.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + 0.477T + 43T^{2} \)
47 \( 1 + (1.22 + 2.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.58 + 6.20i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.26 + 3.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.66 + 9.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.38 - 7.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + (0.347 - 0.602i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.99 - 3.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.77T + 83T^{2} \)
89 \( 1 + (0.118 + 0.205i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.81T + 97T^{2} \)
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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

−10.29664435507912042002745185714, −9.794795749258227522463135361670, −9.332473298370173511231934560843, −8.010916603814810242436262984592, −6.94709698234359572323000821161, −6.01436914234700134903054546306, −5.33108800665849593758506399082, −4.46961091525455701423169807319, −3.26902100995378665439882702664, −2.24882346042668803998549825463, 0.44267403686974045941366001186, 1.39694629458540188444161227433, 2.91636061502607868946642260350, 4.29294890635574559536989018286, 5.53862081386720260502332360472, 6.03899394399709914963196314936, 7.04968116017694430318357736960, 7.64514740964404090436170169882, 8.689551736150188225111972594317, 9.499502289718698078536104242641

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