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Properties

Label 2-950-95.18-c1-0-18
Degree $2$
Conductor $950$
Sign $0.446 - 0.894i$
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 + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s + 3i·9-s + 2·11-s + (4.24 − 4.24i)13-s + 1.41·14-s − 1.00·16-s + (−3 + 3i)17-s + (−2.12 + 2.12i)18-s + (4.24 + i)19-s + (1.41 + 1.41i)22-s + (−1 − i)23-s + 6·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.377 − 0.377i)7-s + (−0.250 + 0.250i)8-s + i·9-s + 0.603·11-s + (1.17 − 1.17i)13-s + 0.377·14-s − 0.250·16-s + (−0.727 + 0.727i)17-s + (−0.499 + 0.499i)18-s + (0.973 + 0.229i)19-s + (0.301 + 0.301i)22-s + (−0.208 − 0.208i)23-s + 1.17·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.446 - 0.894i$
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: \((2,\ 950,\ (\ :1/2),\ 0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93833 + 1.19841i\)
\(L(\frac12)\) \(\approx\) \(1.93833 + 1.19841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.24 - i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-4.24 + 4.24i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8.48iT - 31T^{2} \)
37 \( 1 + (-4.24 - 4.24i)T + 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (8.48 + 8.48i)T + 67iT^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + (9 + 9i)T + 83iT^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + (4.24 + 4.24i)T + 97iT^{2} \)
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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

−10.47453495126117408397793862778, −9.161951939184174383407841428721, −8.217095982245246452738779843562, −7.78619783504216740247987744881, −6.69472329762435480660431454351, −5.84566808015665426875870241924, −4.97875338710202792318699307782, −4.04032593016334732883283840782, −3.02850197860161588080392427030, −1.44807817494085330301590508190, 1.09224996598398599062437472727, 2.37226409512794249815340943665, 3.68913073477991881696100971166, 4.30845811282412531923279970750, 5.53988068858311242324984151069, 6.36513902269706337491651665572, 7.10996802837178866673510985507, 8.480261320899370882769845778311, 9.267625998531578727949606906722, 9.694250279172984659403977935371

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