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Properties

Label 2-950-25.11-c1-0-18
Degree $2$
Conductor $950$
Sign $0.972 - 0.231i$
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  + (−0.809 + 0.587i)2-s + (0.549 − 1.69i)3-s + (0.309 − 0.951i)4-s + (1.41 + 1.73i)5-s + (0.549 + 1.69i)6-s + 4.16·7-s + (0.309 + 0.951i)8-s + (−0.135 − 0.0984i)9-s + (−2.16 − 0.575i)10-s + (1.36 − 0.988i)11-s + (−1.43 − 1.04i)12-s + (1.73 + 1.26i)13-s + (−3.36 + 2.44i)14-s + (3.71 − 1.43i)15-s + (−0.809 − 0.587i)16-s + (0.966 + 2.97i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.317 − 0.977i)3-s + (0.154 − 0.475i)4-s + (0.630 + 0.776i)5-s + (0.224 + 0.691i)6-s + 1.57·7-s + (0.109 + 0.336i)8-s + (−0.0451 − 0.0328i)9-s + (−0.683 − 0.181i)10-s + (0.410 − 0.298i)11-s + (−0.415 − 0.301i)12-s + (0.481 + 0.349i)13-s + (−0.899 + 0.653i)14-s + (0.958 − 0.369i)15-s + (−0.202 − 0.146i)16-s + (0.234 + 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.972 - 0.231i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.972 - 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85400 + 0.217650i\)
\(L(\frac12)\) \(\approx\) \(1.85400 + 0.217650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-1.41 - 1.73i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.549 + 1.69i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
11 \( 1 + (-1.36 + 0.988i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.73 - 1.26i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.966 - 2.97i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (5.44 - 3.95i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.51 - 7.73i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.869 - 2.67i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.49 + 6.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.55 + 5.49i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.38T + 43T^{2} \)
47 \( 1 + (-1.76 + 5.44i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.525 - 1.61i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.13 - 5.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.08 + 0.788i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.182 + 0.562i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.65 + 8.17i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.88 + 2.09i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.54 + 10.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.67 + 8.23i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (4.93 - 3.58i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.53 + 13.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.17640043506921614446025410820, −8.845602677496563804833177569309, −8.413006248068175004069175308679, −7.42993106807442918894379840802, −6.97949697994482585473820353515, −5.98609974102505400766981153244, −5.12695733998641373740795288319, −3.61264989150465984888790641975, −1.83914279995667695138295188911, −1.65947931989066202517654286334, 1.22235050907250215758703224362, 2.26456755478337295966796935268, 3.82477082949449397235729692585, 4.59350018220445528584916683201, 5.34834732948856087414920064261, 6.64446850824606290429231249617, 8.203845260619748203507873886991, 8.244373761095025813958740174078, 9.358203956024123299590718440031, 9.918908253456286393073894444287

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