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Properties

Label 2-952-7.4-c1-0-30
Degree $2$
Conductor $952$
Sign $-0.952 - 0.304i$
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  + (1.25 − 2.17i)3-s + (−1.89 − 3.28i)5-s + (−2.54 + 0.725i)7-s + (−1.63 − 2.83i)9-s + (2.42 − 4.19i)11-s − 2.44·13-s − 9.50·15-s + (−0.5 + 0.866i)17-s + (3.49 + 6.05i)19-s + (−1.61 + 6.43i)21-s + (−2.10 − 3.64i)23-s + (−4.69 + 8.13i)25-s − 0.699·27-s + 3.62·29-s + (−1.59 + 2.75i)31-s + ⋯
L(s)  = 1  + (0.723 − 1.25i)3-s + (−0.848 − 1.46i)5-s + (−0.961 + 0.274i)7-s + (−0.546 − 0.946i)9-s + (0.730 − 1.26i)11-s − 0.679·13-s − 2.45·15-s + (−0.121 + 0.210i)17-s + (0.802 + 1.38i)19-s + (−0.351 + 1.40i)21-s + (−0.439 − 0.760i)23-s + (−0.939 + 1.62i)25-s − 0.134·27-s + 0.672·29-s + (−0.285 + 0.494i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $-0.952 - 0.304i$
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: \((2,\ 952,\ (\ :1/2),\ -0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171424 + 1.10022i\)
\(L(\frac12)\) \(\approx\) \(0.171424 + 1.10022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.725i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.25 + 2.17i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.89 + 3.28i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.42 + 4.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
19 \( 1 + (-3.49 - 6.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.10 + 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.62T + 29T^{2} \)
31 \( 1 + (1.59 - 2.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.00 + 3.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.63T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (5.66 + 9.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.39 + 4.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.87 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.77 - 8.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.54 + 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.382T + 71T^{2} \)
73 \( 1 + (-0.295 + 0.512i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.39 + 7.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.09T + 83T^{2} \)
89 \( 1 + (0.212 + 0.368i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.02T + 97T^{2} \)
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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

−9.204446044818066893217237250735, −8.504103568043345886999853353600, −8.159983251446582590984627772217, −7.16644310997743172591278419609, −6.28069341654891957818216167723, −5.30699191908216460712744526596, −3.93618586420838965891964236599, −3.12480251231278246967608019453, −1.62686289489694471023092539400, −0.47523556361617211041202899459, 2.61212280635134108114879754614, 3.28696581355459010803789350052, 4.05494691022627982358952163722, 4.88007246350684158851192447110, 6.59634835479212878547893449776, 7.06372316644075736980128647303, 7.891490070680500852597734705244, 9.209379448952735318061749024825, 9.782008881984924649580579088951, 10.15358214314333411615744320811

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