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Properties

Label 2-950-95.18-c1-0-20
Degree $2$
Conductor $950$
Sign $-0.305 + 0.952i$
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  + (−0.707 − 0.707i)2-s + (−2.10 + 2.10i)3-s + 1.00i·4-s + 2.97·6-s + (0.978 − 0.978i)7-s + (0.707 − 0.707i)8-s − 5.82i·9-s + 2.43·11-s + (−2.10 − 2.10i)12-s + (−1.31 + 1.31i)13-s − 1.38·14-s − 1.00·16-s + (−5.11 + 5.11i)17-s + (−4.12 + 4.12i)18-s + (−3.48 − 2.61i)19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.21 + 1.21i)3-s + 0.500i·4-s + 1.21·6-s + (0.369 − 0.369i)7-s + (0.250 − 0.250i)8-s − 1.94i·9-s + 0.733·11-s + (−0.606 − 0.606i)12-s + (−0.364 + 0.364i)13-s − 0.369·14-s − 0.250·16-s + (−1.23 + 1.23i)17-s + (−0.971 + 0.971i)18-s + (−0.800 − 0.599i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154163 - 0.211290i\)
\(L(\frac12)\) \(\approx\) \(0.154163 - 0.211290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
19 \( 1 + (3.48 + 2.61i)T \)
good3 \( 1 + (2.10 - 2.10i)T - 3iT^{2} \)
7 \( 1 + (-0.978 + 0.978i)T - 7iT^{2} \)
11 \( 1 - 2.43T + 11T^{2} \)
13 \( 1 + (1.31 - 1.31i)T - 13iT^{2} \)
17 \( 1 + (5.11 - 5.11i)T - 17iT^{2} \)
23 \( 1 + (1.29 + 1.29i)T + 23iT^{2} \)
29 \( 1 - 0.448T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (4.60 + 4.60i)T + 37iT^{2} \)
41 \( 1 + 6.28iT - 41T^{2} \)
43 \( 1 + (7.27 + 7.27i)T + 43iT^{2} \)
47 \( 1 + (-6.15 + 6.15i)T - 47iT^{2} \)
53 \( 1 + (-9.37 + 9.37i)T - 53iT^{2} \)
59 \( 1 + 5.89T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-4.17 - 4.17i)T + 67iT^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (-8.04 - 8.04i)T + 73iT^{2} \)
79 \( 1 + 6.23T + 79T^{2} \)
83 \( 1 + (11.7 + 11.7i)T + 83iT^{2} \)
89 \( 1 - 3.31T + 89T^{2} \)
97 \( 1 + (-6.78 - 6.78i)T + 97iT^{2} \)
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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

−10.13742145699237213704475394361, −9.019101408469098673323526948003, −8.622167020520386001720723931167, −7.04261130240457547890909713607, −6.38548231871956463875078268662, −5.22875052972259987403349883380, −4.27128853218898473289922738570, −3.81644734051729413618308417396, −1.95186117526529788187082794872, −0.18268171974051934572528242049, 1.24549120850048610478693420565, 2.37683434450932545442520746077, 4.50899906556251965541867149694, 5.37905363512299851562925423409, 6.26077186375292680267963088238, 6.79045348413923735098585603233, 7.61168567464117108767909520495, 8.396045636101028796381181593357, 9.360037932985818604890192835456, 10.37913066623677419390586347595

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