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Properties

Label 2-950-95.44-c1-0-1
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  + (−0.642 − 0.766i)2-s + (−0.460 + 1.26i)3-s + (−0.173 + 0.984i)4-s + (1.26 − 0.460i)6-s + (−2.49 − 1.43i)7-s + (0.866 − 0.500i)8-s + (0.907 + 0.761i)9-s + (0.5 + 0.866i)11-s + (−1.16 − 0.673i)12-s + (0.181 + 0.5i)13-s + (0.500 + 2.83i)14-s + (−0.939 − 0.342i)16-s + (−1.06 − 1.26i)17-s − 1.18i·18-s + (−3.79 + 2.15i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (−0.266 + 0.730i)3-s + (−0.0868 + 0.492i)4-s + (0.516 − 0.188i)6-s + (−0.942 − 0.544i)7-s + (0.306 − 0.176i)8-s + (0.302 + 0.253i)9-s + (0.150 + 0.261i)11-s + (−0.336 − 0.194i)12-s + (0.0504 + 0.138i)13-s + (0.133 + 0.757i)14-s + (−0.234 − 0.0855i)16-s + (−0.257 − 0.307i)17-s − 0.279i·18-s + (−0.869 + 0.493i)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: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00833228 + 0.121363i\)
\(L(\frac12)\) \(\approx\) \(0.00833228 + 0.121363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (3.79 - 2.15i)T \)
good3 \( 1 + (0.460 - 1.26i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.49 + 1.43i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.181 - 0.5i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.06 + 1.26i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.10 + 0.194i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.81 - 1.52i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.847 + 1.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.59iT - 37T^{2} \)
41 \( 1 + (4.85 + 1.76i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (10.6 - 1.88i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.16 + 1.39i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (11.8 + 2.08i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (6.72 - 5.64i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.08 + 6.15i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.62 + 1.93i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.02 + 5.79i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.70 - 4.68i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (8.05 + 2.93i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (9.22 + 5.32i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.694 + 0.252i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-7.45 - 8.88i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.27199367384461019040992520953, −9.859626365609504954873534815454, −9.070282126517125211649562122510, −8.070900846559973478868332067275, −7.07199270312745611544581405153, −6.28527124782307443983752197772, −4.92307128596710977437811867516, −4.10102643463251371003861456432, −3.25108570942734873162856720623, −1.80476256065539384115494170827, 0.06700602036081832376572520114, 1.62809793481435864157670998924, 3.04276791054192700907611487544, 4.40159137922022164576665049889, 5.68569880205782378588431242120, 6.50277530263848781276417486061, 6.79825195129039910956740812248, 7.981233389152325809133681758909, 8.711684696764151353628149652569, 9.568034895377964994471774439058

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