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Properties

Label 2-950-25.21-c1-0-32
Degree $2$
Conductor $950$
Sign $0.977 - 0.211i$
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  + (−0.309 + 0.951i)2-s + (0.823 + 0.598i)3-s + (−0.809 − 0.587i)4-s + (−1.34 + 1.78i)5-s + (−0.823 + 0.598i)6-s + 3.63·7-s + (0.809 − 0.587i)8-s + (−0.607 − 1.86i)9-s + (−1.27 − 1.83i)10-s + (1.67 − 5.14i)11-s + (−0.314 − 0.967i)12-s + (−1.35 − 4.17i)13-s + (−1.12 + 3.46i)14-s + (−2.17 + 0.661i)15-s + (0.309 + 0.951i)16-s + (3.52 − 2.56i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.475 + 0.345i)3-s + (−0.404 − 0.293i)4-s + (−0.603 + 0.797i)5-s + (−0.336 + 0.244i)6-s + 1.37·7-s + (0.286 − 0.207i)8-s + (−0.202 − 0.622i)9-s + (−0.404 − 0.580i)10-s + (0.504 − 1.55i)11-s + (−0.0907 − 0.279i)12-s + (−0.376 − 1.15i)13-s + (−0.300 + 0.924i)14-s + (−0.562 + 0.170i)15-s + (0.0772 + 0.237i)16-s + (0.854 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59240 + 0.169919i\)
\(L(\frac12)\) \(\approx\) \(1.59240 + 0.169919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (1.34 - 1.78i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.823 - 0.598i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 3.63T + 7T^{2} \)
11 \( 1 + (-1.67 + 5.14i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.35 + 4.17i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.52 + 2.56i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-1.10 + 3.39i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.73 - 3.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (7.44 - 5.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.35 - 4.15i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.42 + 7.47i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.71T + 43T^{2} \)
47 \( 1 + (2.86 + 2.07i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.29 - 4.57i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.12 - 12.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.42 - 4.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.47 + 1.07i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.0774 + 0.0562i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.47 - 4.53i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.78 - 4.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.34 + 5.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.84 + 14.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.41 - 1.03i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.14380268759406418037509295295, −8.753251912210313067031564372161, −8.515826389462418209601519479325, −7.65212444890101162963306266636, −6.84090767266158682031853011302, −5.74711605277945457875968309272, −4.91709040932620887437718410491, −3.62834437712240679549397306814, −2.96233624090080381122803843125, −0.846063355157717292424584778196, 1.58169908284350895766075217839, 1.99090521512850477837885256988, 3.75305940328737311771832276252, 4.63694885873819610865180208342, 5.19442074092713320754277084957, 7.01331493657900914929591035724, 7.905974059243231275745118165538, 8.146470181521245334310003546415, 9.248150524787288295082279930017, 9.815651395777754849208377834316

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