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Properties

Label 2-950-95.12-c1-0-4
Degree $2$
Conductor $950$
Sign $-0.985 - 0.169i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.965 − 0.258i)2-s + (−0.431 + 1.61i)3-s + (0.866 + 0.499i)4-s + (0.834 − 1.44i)6-s + (2.81 + 2.81i)7-s + (−0.707 − 0.707i)8-s + (0.185 + 0.107i)9-s − 5.57·11-s + (−1.18 + 1.18i)12-s + (−3.10 + 0.831i)13-s + (−1.98 − 3.44i)14-s + (0.500 + 0.866i)16-s + (−1.06 + 3.97i)17-s + (−0.151 − 0.151i)18-s + (−0.254 − 4.35i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.249 + 0.930i)3-s + (0.433 + 0.249i)4-s + (0.340 − 0.590i)6-s + (1.06 + 1.06i)7-s + (−0.249 − 0.249i)8-s + (0.0618 + 0.0356i)9-s − 1.68·11-s + (−0.340 + 0.340i)12-s + (−0.861 + 0.230i)13-s + (−0.531 − 0.919i)14-s + (0.125 + 0.216i)16-s + (−0.258 + 0.963i)17-s + (−0.0356 − 0.0356i)18-s + (−0.0584 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0551824 + 0.645699i\)
\(L(\frac12)\) \(\approx\) \(0.0551824 + 0.645699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (0.254 + 4.35i)T \)
good3 \( 1 + (0.431 - 1.61i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.81 - 2.81i)T + 7iT^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 + (3.10 - 0.831i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.06 - 3.97i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.02 - 3.83i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.09 + 3.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.573iT - 31T^{2} \)
37 \( 1 + (-7.24 + 7.24i)T - 37iT^{2} \)
41 \( 1 + (5.12 - 2.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.37 - 1.44i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (11.7 - 3.15i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.66 - 0.981i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.75 - 6.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.06 - 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.07 + 11.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.56 - 2.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.0 + 3.50i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.48 - 4.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.30 - 7.30i)T - 83iT^{2} \)
89 \( 1 + (-7.08 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.31 + 0.619i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.40174535861759374850008768174, −9.663851676413516733799721242751, −8.899343103640717326333970457905, −8.009377429650880276099592461950, −7.43932354080061683091679087456, −5.94465583873092763351291626175, −5.07492157908468207760967888261, −4.48286103477622391060218538517, −2.84470163956216194368675849806, −1.95123721532475929045861523107, 0.37747227600096353541294293495, 1.60869846078208060692162649712, 2.77241007528662208990906588603, 4.55503532546562089096687698124, 5.31222810437463326914857086465, 6.56569019006659992602297707045, 7.37537532433242919535130482088, 7.77994009091143459727862193290, 8.444740739753547864456991267532, 9.923270242944900975040027792605

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