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Properties

Label 2-952-952.373-c1-0-119
Degree $2$
Conductor $952$
Sign $0.00865 - 0.999i$
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 + (2.70 − 3.11i)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 + (0.655 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00865 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00865 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.00865 - 0.999i$
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: \((2,\ 952,\ (\ :1/2),\ 0.00865 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0641209 + 0.0635684i\)
\(L(\frac12)\) \(\approx\) \(0.0641209 + 0.0635684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.70 + 3.11i)T \)
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

−9.589870681504665335232719321380, −8.175024661747981058064067493412, −7.80478215190891063432524434113, −6.98064032600933313300785908693, −6.22339748635604706885211856548, −5.56739798307423201836523606018, −3.32427288526125420584001736447, −2.80313274378377868098730433935, −1.00515575629534757640672282787, −0.07682120841232603885408774551, 1.97037938617353548293284333184, 3.61825203662245790312876004119, 4.60940103779819297319164458639, 5.53102226732852576503755108066, 6.40515664282793873970476972578, 7.31732591801210243468869775585, 8.395637925783693125054110766391, 9.304818435824020660910144067746, 9.790112016443055915019880485997, 10.31042374591766033452115872580

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