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Properties

Label 2-950-475.113-c1-0-21
Degree $2$
Conductor $950$
Sign $-0.818 - 0.574i$
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  + (0.156 + 0.987i)2-s + (2.11 + 1.07i)3-s + (−0.951 + 0.309i)4-s + (0.421 + 2.19i)5-s + (−0.734 + 2.26i)6-s + (0.281 − 0.281i)7-s + (−0.453 − 0.891i)8-s + (1.55 + 2.14i)9-s + (−2.10 + 0.759i)10-s + (0.297 + 0.216i)11-s + (−2.34 − 0.371i)12-s + (−1.00 + 6.33i)13-s + (0.322 + 0.234i)14-s + (−1.47 + 5.10i)15-s + (0.809 − 0.587i)16-s + (−1.24 − 2.43i)17-s + ⋯
L(s)  = 1  + (0.110 + 0.698i)2-s + (1.22 + 0.622i)3-s + (−0.475 + 0.154i)4-s + (0.188 + 0.982i)5-s + (−0.299 + 0.922i)6-s + (0.106 − 0.106i)7-s + (−0.160 − 0.315i)8-s + (0.518 + 0.713i)9-s + (−0.665 + 0.240i)10-s + (0.0897 + 0.0651i)11-s + (−0.677 − 0.107i)12-s + (−0.278 + 1.75i)13-s + (0.0861 + 0.0625i)14-s + (−0.381 + 1.31i)15-s + (0.202 − 0.146i)16-s + (−0.301 − 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.818 - 0.574i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711008 + 2.24903i\)
\(L(\frac12)\) \(\approx\) \(0.711008 + 2.24903i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
5 \( 1 + (-0.421 - 2.19i)T \)
19 \( 1 + (-0.349 + 4.34i)T \)
good3 \( 1 + (-2.11 - 1.07i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-0.281 + 0.281i)T - 7iT^{2} \)
11 \( 1 + (-0.297 - 0.216i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.00 - 6.33i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.24 + 2.43i)T + (-9.99 + 13.7i)T^{2} \)
23 \( 1 + (-0.556 - 3.51i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-0.0977 - 0.300i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.98 - 1.62i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.49 - 0.870i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-3.26 - 4.49i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (7.94 + 7.94i)T + 43iT^{2} \)
47 \( 1 + (-2.98 - 1.52i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (11.6 + 5.95i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-12.2 + 8.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.23 - 3.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.80 + 2.44i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (-3.12 + 1.01i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.297 + 1.87i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.91 + 5.88i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.6 - 5.40i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (-9.24 - 6.71i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.41 + 6.70i)T + (-57.0 - 78.4i)T^{2} \)
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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

−9.895529105713980386163813623101, −9.486959231517809424604826805500, −8.795374929388504152167168129337, −7.83167307767139929641959259461, −6.97569866696611277569557498316, −6.42178518733823788707140650404, −4.93393375627590380213426659738, −4.12907640910706998637438509302, −3.15967372254554630596941442071, −2.19224082723970244846202283677, 0.957796653775926615906657426247, 2.11837026105193262362568611094, 3.02869765963827903522785120879, 4.08939705309168469212395300157, 5.21027497176571189461989896120, 6.12540729364052946980895322029, 7.63053274739641731975326956313, 8.286422901739368951041600214000, 8.646948381196588030398960338821, 9.740105536123680200630953526192

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