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Properties

Label 2-950-95.64-c1-0-10
Degree $2$
Conductor $950$
Sign $0.629 - 0.776i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + 5·11-s + (−1.73 − i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 3i·18-s + (0.5 + 4.33i)19-s + (−4.33 + 2.5i)22-s + (−0.866 − 0.5i)23-s + 1.99·26-s + (−0.866 − 0.499i)28-s + (3 − 5.19i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.377i·7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + 1.50·11-s + (−0.480 − 0.277i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.707i·18-s + (0.114 + 0.993i)19-s + (−0.923 + 0.533i)22-s + (−0.180 − 0.104i)23-s + 0.392·26-s + (−0.163 − 0.0944i)28-s + (0.557 − 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.629 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05036 + 0.500503i\)
\(L(\frac12)\) \(\approx\) \(1.05036 + 0.500503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 - 4.33i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 11iT - 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.33 + 2.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12.1 + 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.991636505283153600898452516371, −9.411242852936490387552439736293, −8.264143916443301440426502366594, −7.897542779043087583682892340349, −6.74943269485541353215531228298, −6.09594088397369968938734196635, −4.98413533179454409611826794399, −3.94771064800355390066171331379, −2.53035068755269575729743950047, −1.15903865807132168525983540225, 0.817236759654724690694854495303, 2.28682691439469591848265783750, 3.41958227813304733817770232648, 4.39524513712003309553668276742, 5.76971478482611123558090108084, 6.65631233052467521579268602621, 7.34645589792101254308892114552, 8.656371100296621739193766313132, 9.114766174517066374772010586020, 9.637035126756770122585321978624

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