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Properties

Label 2-950-95.12-c1-0-2
Degree $2$
Conductor $950$
Sign $-0.989 + 0.147i$
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 + (−0.407 + 1.52i)3-s + (0.866 + 0.499i)4-s + (−0.788 + 1.36i)6-s + (−2.78 − 2.78i)7-s + (0.707 + 0.707i)8-s + (0.446 + 0.257i)9-s − 3.61·11-s + (−1.11 + 1.11i)12-s + (−4.02 + 1.07i)13-s + (−1.97 − 3.41i)14-s + (0.500 + 0.866i)16-s + (−1.78 + 6.67i)17-s + (0.364 + 0.364i)18-s + (−4.35 + 0.0613i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.235 + 0.879i)3-s + (0.433 + 0.249i)4-s + (−0.321 + 0.557i)6-s + (−1.05 − 1.05i)7-s + (0.249 + 0.249i)8-s + (0.148 + 0.0858i)9-s − 1.08·11-s + (−0.321 + 0.321i)12-s + (−1.11 + 0.299i)13-s + (−0.526 − 0.912i)14-s + (0.125 + 0.216i)16-s + (−0.434 + 1.61i)17-s + (0.0858 + 0.0858i)18-s + (−0.999 + 0.0140i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.989 + 0.147i$
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: \((2,\ 950,\ (\ :1/2),\ -0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0515814 - 0.695796i\)
\(L(\frac12)\) \(\approx\) \(0.0515814 - 0.695796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.35 - 0.0613i)T \)
good3 \( 1 + (0.407 - 1.52i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.78 + 2.78i)T + 7iT^{2} \)
11 \( 1 + 3.61T + 11T^{2} \)
13 \( 1 + (4.02 - 1.07i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.78 - 6.67i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.826 + 3.08i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.966 - 1.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.38iT - 31T^{2} \)
37 \( 1 + (5.07 - 5.07i)T - 37iT^{2} \)
41 \( 1 + (-1.72 + 0.994i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.85 - 2.10i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.79 + 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.1 + 3.25i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.99 + 8.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.12 + 1.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.12 + 4.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.67 + 2.69i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.2 + 3.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.484 + 0.839i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.90 - 4.90i)T - 83iT^{2} \)
89 \( 1 + (0.366 - 0.635i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.9 - 3.20i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.39629237585970291191179677168, −10.09711625248938888633721685591, −8.884345451545832603690934436596, −7.75439150386630092471538167373, −6.94892067527189065768750953260, −6.14652868261392589075302889240, −5.02913336063352257100316699104, −4.28252369284336369902289101360, −3.59332843794277401285492700562, −2.27007071254469426258194731799, 0.23822204967147018056630329774, 2.32792263402885943459823371113, 2.74438027496775604146337883511, 4.27984453557534463341862736171, 5.47928637536390111841203965623, 5.95427210322361140881459424302, 7.14461179144766349794276993942, 7.42752718662485314057030367074, 8.870847872635890448851358365427, 9.679755446389134784287827860092

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