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Properties

Label 2-950-95.37-c1-0-18
Degree $2$
Conductor $950$
Sign $-0.990 - 0.136i$
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  + (−0.707 + 0.707i)2-s + (−1.68 − 1.68i)3-s − 1.00i·4-s + 2.38·6-s + (2.49 + 2.49i)7-s + (0.707 + 0.707i)8-s + 2.68i·9-s − 3.60·11-s + (−1.68 + 1.68i)12-s + (−2.82 − 2.82i)13-s − 3.52·14-s − 1.00·16-s + (2.34 + 2.34i)17-s + (−1.89 − 1.89i)18-s + (1.57 − 4.06i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.973 − 0.973i)3-s − 0.500i·4-s + 0.973·6-s + (0.942 + 0.942i)7-s + (0.250 + 0.250i)8-s + 0.894i·9-s − 1.08·11-s + (−0.486 + 0.486i)12-s + (−0.782 − 0.782i)13-s − 0.942·14-s − 0.250·16-s + (0.567 + 0.567i)17-s + (−0.447 − 0.447i)18-s + (0.360 − 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00194464 + 0.0283413i\)
\(L(\frac12)\) \(\approx\) \(0.00194464 + 0.0283413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 + (-1.57 + 4.06i)T \)
good3 \( 1 + (1.68 + 1.68i)T + 3iT^{2} \)
7 \( 1 + (-2.49 - 2.49i)T + 7iT^{2} \)
11 \( 1 + 3.60T + 11T^{2} \)
13 \( 1 + (2.82 + 2.82i)T + 13iT^{2} \)
17 \( 1 + (-2.34 - 2.34i)T + 17iT^{2} \)
23 \( 1 + (0.793 - 0.793i)T - 23iT^{2} \)
29 \( 1 + 4.81T + 29T^{2} \)
31 \( 1 + 3.69iT - 31T^{2} \)
37 \( 1 + (-2.34 + 2.34i)T - 37iT^{2} \)
41 \( 1 - 7.28iT - 41T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - 43iT^{2} \)
47 \( 1 + (7.44 + 7.44i)T + 47iT^{2} \)
53 \( 1 + (2.44 + 2.44i)T + 53iT^{2} \)
59 \( 1 + 8.96T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (8.19 - 8.19i)T - 67iT^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + (7.33 - 7.33i)T - 73iT^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + (1.38 - 1.38i)T - 83iT^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-3.46 + 3.46i)T - 97iT^{2} \)
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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

−9.595724928700144700962288119140, −8.471548295444040992279092168217, −7.75090561212782599734579574515, −7.27097758333958927083942378206, −6.03397803148660876930923253039, −5.50232504712462511484190256130, −4.85282105911109800153877234829, −2.70111894983246361049943108295, −1.53480384106916663953547266484, −0.01783801840068589784007142120, 1.68604418602471697495709871471, 3.28530886431721926335029494048, 4.52445490364977429741061855111, 4.90080636324811757984158070475, 6.00664514708845823548929541363, 7.49819245291888566141802837804, 7.76327837141400319119051347174, 9.143622702697257786913050832979, 9.936456505296209024132803348459, 10.53265905641719627961307268373

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