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Properties

Label 2-950-95.18-c1-0-19
Degree $2$
Conductor $950$
Sign $0.759 + 0.650i$
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 + (−0.707 + 0.707i)3-s + 1.00i·4-s + 1.00·6-s + (2.73 − 2.73i)7-s + (0.707 − 0.707i)8-s + 1.99i·9-s + (−0.707 − 0.707i)12-s + (0.707 − 0.707i)13-s − 3.87·14-s − 1.00·16-s + (2.73 − 2.73i)17-s + (1.41 − 1.41i)18-s + (−3.87 − 2i)19-s + 3.87i·21-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + 0.408·6-s + (1.03 − 1.03i)7-s + (0.250 − 0.250i)8-s + 0.666i·9-s + (−0.204 − 0.204i)12-s + (0.196 − 0.196i)13-s − 1.03·14-s − 0.250·16-s + (0.664 − 0.664i)17-s + (0.333 − 0.333i)18-s + (−0.888 − 0.458i)19-s + 0.845i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.759 + 0.650i$
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: \((2,\ 950,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10672 - 0.409328i\)
\(L(\frac12)\) \(\approx\) \(1.10672 - 0.409328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.87 + 2i)T \)
good3 \( 1 + (0.707 - 0.707i)T - 3iT^{2} \)
7 \( 1 + (-2.73 + 2.73i)T - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \)
17 \( 1 + (-2.73 + 2.73i)T - 17iT^{2} \)
23 \( 1 + (-2.73 - 2.73i)T + 23iT^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + 37iT^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-5.47 + 5.47i)T - 47iT^{2} \)
53 \( 1 + (-6.36 + 6.36i)T - 53iT^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (9.19 + 9.19i)T + 67iT^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (2.73 + 2.73i)T + 73iT^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-5.47 - 5.47i)T + 83iT^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + (-5.65 - 5.65i)T + 97iT^{2} \)
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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

−10.30042489685193486754896540351, −9.239242899384424303856610673682, −8.213707831476766140667852723731, −7.65664114990535511496113153918, −6.73948936944275147149023126516, −5.27996846013383771615769632748, −4.64616092224960490087168494339, −3.64909703253174139461402082186, −2.22103519689679304277763265989, −0.872906020208724217421846810586, 1.13608758998098718595454161644, 2.32944653671989894751878500588, 4.00386566053095127087883551686, 5.23623282179349420854830818158, 5.92433401289976454217290021867, 6.68020067482736616899931726498, 7.66119973828738473059604862694, 8.611769818399855152658659211264, 8.902130174631104185676313649270, 10.14799933985077336652682159208

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