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Properties

Label 2-950-19.4-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.444 + 0.895i$
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  + (−0.939 − 0.342i)2-s + (0.558 + 3.16i)3-s + (0.766 + 0.642i)4-s + (0.558 − 3.16i)6-s + (−0.0116 + 0.0202i)7-s + (−0.500 − 0.866i)8-s + (−6.89 + 2.51i)9-s + (−1.08 − 1.87i)11-s + (−1.60 + 2.78i)12-s + (0.276 − 1.56i)13-s + (0.0179 − 0.0150i)14-s + (0.173 + 0.984i)16-s + (−7.49 − 2.72i)17-s + 7.34·18-s + (1.06 − 4.22i)19-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.322 + 1.82i)3-s + (0.383 + 0.321i)4-s + (0.227 − 1.29i)6-s + (−0.00442 + 0.00765i)7-s + (−0.176 − 0.306i)8-s + (−2.29 + 0.836i)9-s + (−0.325 − 0.564i)11-s + (−0.464 + 0.803i)12-s + (0.0765 − 0.434i)13-s + (0.00478 − 0.00401i)14-s + (0.0434 + 0.246i)16-s + (−1.81 − 0.661i)17-s + 1.73·18-s + (0.244 − 0.969i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338095 - 0.0544967i\)
\(L(\frac12)\) \(\approx\) \(0.0338095 - 0.0544967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-1.06 + 4.22i)T \)
good3 \( 1 + (-0.558 - 3.16i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (0.0116 - 0.0202i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.276 + 1.56i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (7.49 + 2.72i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (3.43 + 2.88i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.09 - 2.58i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.22 - 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.389T + 37T^{2} \)
41 \( 1 + (-0.972 - 5.51i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.43 - 6.23i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-4.01 + 1.46i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-2.89 - 2.43i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-3.41 - 1.24i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.84 + 7.41i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.62 + 1.31i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.85 - 8.27i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.330 - 1.87i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (1.65 + 9.36i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.31 - 2.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.54 + 8.74i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-10.4 - 3.81i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.56045110730129937847263271455, −9.726909852475450918477164318315, −8.966231130403111999092560449838, −8.644353373091390381035428953259, −7.52777113950596113796351175634, −6.25086855940949925597508120776, −5.12968861647777958975774804208, −4.36215950196678014293529816548, −3.28505444247886721906113173257, −2.47955038437777215163755265363, 0.03242759227656129028630490468, 1.78251736259043406477146676129, 2.20399639090051453552833420732, 3.86096785488321356496166037193, 5.61514661466984571388550164366, 6.31549044733053755637123937264, 7.17903921171019671859920278168, 7.63944424447121171875014374287, 8.525713320964107159473421162917, 9.108357664071649365181923668474

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