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Properties

Label 2-950-25.21-c1-0-20
Degree $2$
Conductor $950$
Sign $-0.567 + 0.823i$
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  + (0.309 − 0.951i)2-s + (−2.41 − 1.75i)3-s + (−0.809 − 0.587i)4-s + (−0.597 + 2.15i)5-s + (−2.41 + 1.75i)6-s + 1.43·7-s + (−0.809 + 0.587i)8-s + (1.83 + 5.64i)9-s + (1.86 + 1.23i)10-s + (0.998 − 3.07i)11-s + (0.923 + 2.84i)12-s + (0.816 + 2.51i)13-s + (0.442 − 1.36i)14-s + (5.23 − 4.16i)15-s + (0.309 + 0.951i)16-s + (1.16 − 0.843i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−1.39 − 1.01i)3-s + (−0.404 − 0.293i)4-s + (−0.266 + 0.963i)5-s + (−0.987 + 0.717i)6-s + 0.541·7-s + (−0.286 + 0.207i)8-s + (0.611 + 1.88i)9-s + (0.589 + 0.390i)10-s + (0.301 − 0.926i)11-s + (0.266 + 0.820i)12-s + (0.226 + 0.696i)13-s + (0.118 − 0.363i)14-s + (1.35 − 1.07i)15-s + (0.0772 + 0.237i)16-s + (0.281 − 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446218 - 0.849385i\)
\(L(\frac12)\) \(\approx\) \(0.446218 - 0.849385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.597 - 2.15i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (2.41 + 1.75i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 1.43T + 7T^{2} \)
11 \( 1 + (-0.998 + 3.07i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.816 - 2.51i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.16 + 0.843i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-2.18 + 6.71i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.77 - 5.64i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.326 - 0.237i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.89 + 8.92i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.706 - 2.17i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.686T + 43T^{2} \)
47 \( 1 + (9.36 + 6.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.21 - 3.78i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.57 - 11.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.46 + 7.57i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-10.8 + 7.85i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (13.1 + 9.57i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.81 + 11.7i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.41 - 3.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.70 + 1.96i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.97 - 6.08i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.19 + 3.77i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.32602045822669457294834953932, −8.892811135539185884855008563118, −7.936612549114086323881288375454, −6.84921887452882720914755040009, −6.44157647388542855114887890527, −5.44671669284806729231537761517, −4.52474071231803505663088905440, −3.18767959749225116631487300083, −1.90428948233131674006898919323, −0.63879755024723563392110136192, 1.10195078010447671255798393454, 3.63544128819169278189787414928, 4.54792952682749833048650307173, 5.06861692902558570823636730160, 5.72736057933966731995404929028, 6.70460627129023566027266356652, 7.81384719001429228993808825124, 8.606481622071847473038913145394, 9.789925318127403829923085235652, 10.02706661789354826181192837914

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