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Properties

Label 2-950-95.54-c1-0-15
Degree $2$
Conductor $950$
Sign $0.0605 + 0.998i$
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 + (−0.524 − 1.43i)3-s + (−0.173 − 0.984i)4-s + (1.43 + 0.524i)6-s + (−0.601 + 0.347i)7-s + (0.866 + 0.500i)8-s + (0.5 − 0.419i)9-s + (1.06 − 1.83i)11-s + (−1.32 + 0.766i)12-s + (−0.363 + i)13-s + (0.120 − 0.684i)14-s + (−0.939 + 0.342i)16-s + (0.196 − 0.233i)17-s + 0.652i·18-s + (4.11 − 1.43i)19-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.302 − 0.831i)3-s + (−0.0868 − 0.492i)4-s + (0.587 + 0.213i)6-s + (−0.227 + 0.131i)7-s + (0.306 + 0.176i)8-s + (0.166 − 0.139i)9-s + (0.319 − 0.553i)11-s + (−0.383 + 0.221i)12-s + (−0.100 + 0.277i)13-s + (0.0322 − 0.182i)14-s + (−0.234 + 0.0855i)16-s + (0.0476 − 0.0567i)17-s + 0.153i·18-s + (0.944 − 0.328i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0605 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0605 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.0605 + 0.998i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.0605 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685416 - 0.645087i\)
\(L(\frac12)\) \(\approx\) \(0.685416 - 0.645087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 + 1.43i)T \)
good3 \( 1 + (0.524 + 1.43i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.601 - 0.347i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.06 + 1.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.363 - i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.196 + 0.233i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.921 - 0.162i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.467 + 0.392i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.41 + 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (-7.39 + 2.68i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.664 - 0.117i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (3.64 + 4.34i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (2.65 - 0.467i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (7.56 + 6.35i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.16 + 6.59i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.15 + 9.71i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.12 + 12.0i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.37 + 6.52i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.30 - 0.839i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-15.0 + 8.69i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (15.0 + 5.48i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.00 - 4.77i)T + (-16.8 - 95.5i)T^{2} \)
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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

−9.499666307087980639983855550553, −9.135578377162685207104875365632, −7.87633109890735730833402626499, −7.38436824565356017486024682601, −6.40772966737603371545988712157, −5.93131161777834308457283044568, −4.74122216154985181966706054650, −3.40832824136351782363491183194, −1.85900792155926311385114023076, −0.57674557407785002498450668483, 1.42117689252060511622203872860, 2.91164620313259644991255862477, 3.96951046625962818518761518982, 4.76567753138093589134090774711, 5.78470239730035871782037680083, 7.04251958692534956905125736792, 7.78412528383125756453609538816, 8.839912865821267183202420397610, 9.802752010200513166701020271272, 10.00276201980421908506661750081

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