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Properties

Label 2-950-19.5-c1-0-30
Degree $2$
Conductor $950$
Sign $-0.997 + 0.0667i$
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  + (0.939 − 0.342i)2-s + (0.443 − 2.51i)3-s + (0.766 − 0.642i)4-s + (−0.443 − 2.51i)6-s + (−2.28 − 3.95i)7-s + (0.500 − 0.866i)8-s + (−3.30 − 1.20i)9-s + (−1.71 + 2.96i)11-s + (−1.27 − 2.21i)12-s + (−0.141 − 0.803i)13-s + (−3.49 − 2.93i)14-s + (0.173 − 0.984i)16-s + (−3.99 + 1.45i)17-s − 3.52·18-s + (4.31 + 0.637i)19-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.256 − 1.45i)3-s + (0.383 − 0.321i)4-s + (−0.181 − 1.02i)6-s + (−0.862 − 1.49i)7-s + (0.176 − 0.306i)8-s + (−1.10 − 0.401i)9-s + (−0.516 + 0.894i)11-s + (−0.368 − 0.638i)12-s + (−0.0392 − 0.222i)13-s + (−0.934 − 0.784i)14-s + (0.0434 − 0.246i)16-s + (−0.968 + 0.352i)17-s − 0.830·18-s + (0.989 + 0.146i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 + 0.0667i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.997 + 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0652294 - 1.95258i\)
\(L(\frac12)\) \(\approx\) \(0.0652294 - 1.95258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-4.31 - 0.637i)T \)
good3 \( 1 + (-0.443 + 2.51i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.28 + 3.95i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.71 - 2.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.141 + 0.803i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.99 - 1.45i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.70 + 3.10i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (2.25 + 0.820i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.45 - 5.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.82T + 37T^{2} \)
41 \( 1 + (-2.12 + 12.0i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.38 - 2.83i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (6.98 + 2.54i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-8.64 + 7.25i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.29 + 1.92i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-7.36 + 6.17i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.29 - 1.92i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (11.7 + 9.87i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.61 - 9.16i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.376 - 2.13i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.240 - 0.416i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.474 + 2.68i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.87 - 0.682i)T + (74.3 - 62.3i)T^{2} \)
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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

−9.901468785946107501125457408475, −8.617681637997181594224767143669, −7.52307928572415263108836668213, −6.99810396253199026467491159396, −6.57135889154769883195722372798, −5.25382847616591275893704103104, −4.11444883312006474550168465660, −3.07314599198034882461212645141, −1.97454159985370242242808960644, −0.68818204161373129559435594714, 2.68020751549417411180896289886, 3.15581478409444652724844644300, 4.26626969717581907819287856932, 5.29256943318653602110603085505, 5.75282372580829575702743516999, 6.82282380370749968003033199293, 8.141133496235615824011083136393, 9.093001652660735087008451921787, 9.381748792241205270397425848281, 10.38186819713078314343857650364

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