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Properties

Label 2-950-95.54-c1-0-18
Degree $2$
Conductor $950$
Sign $0.703 + 0.710i$
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  + (0.642 − 0.766i)2-s + (0.460 + 1.26i)3-s + (−0.173 − 0.984i)4-s + (1.26 + 0.460i)6-s + (2.49 − 1.43i)7-s + (−0.866 − 0.500i)8-s + (0.907 − 0.761i)9-s + (0.5 − 0.866i)11-s + (1.16 − 0.673i)12-s + (−0.181 + 0.5i)13-s + (0.500 − 2.83i)14-s + (−0.939 + 0.342i)16-s + (1.06 − 1.26i)17-s − 1.18i·18-s + (−3.79 − 2.15i)19-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.266 + 0.730i)3-s + (−0.0868 − 0.492i)4-s + (0.516 + 0.188i)6-s + (0.942 − 0.544i)7-s + (−0.306 − 0.176i)8-s + (0.302 − 0.253i)9-s + (0.150 − 0.261i)11-s + (0.336 − 0.194i)12-s + (−0.0504 + 0.138i)13-s + (0.133 − 0.757i)14-s + (−0.234 + 0.0855i)16-s + (0.257 − 0.307i)17-s − 0.279i·18-s + (−0.869 − 0.493i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27998 - 0.950820i\)
\(L(\frac12)\) \(\approx\) \(2.27998 - 0.950820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (3.79 + 2.15i)T \)
good3 \( 1 + (-0.460 - 1.26i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-2.49 + 1.43i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.181 - 0.5i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.06 + 1.26i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.10 + 0.194i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.81 + 1.52i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.847 - 1.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.59iT - 37T^{2} \)
41 \( 1 + (4.85 - 1.76i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-10.6 - 1.88i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.16 + 1.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-11.8 + 2.08i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (6.72 + 5.64i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.08 - 6.15i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.62 + 1.93i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.02 - 5.79i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-1.70 - 4.68i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (8.05 - 2.93i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-9.22 + 5.32i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.694 - 0.252i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.45 - 8.88i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.12218542250624030690602106259, −9.223979013030928375782388859022, −8.503149201506738493825515930289, −7.39582647757251393214223040390, −6.43784672109760117356613968320, −5.19857467109024239872240490299, −4.42670313657126617490007157648, −3.79445608237762631145620705069, −2.56999997241900737612565992072, −1.13339199569155031815894476811, 1.57918230874872473749685596578, 2.58907987920807839225095204612, 4.06919077848064316650410403564, 4.94336524271539219725062349248, 5.87741557637933044679092204473, 6.80727899779037069679968567687, 7.63611945916229408619877295178, 8.244283191365496781310968089473, 8.942665102616883051091621326259, 10.19694022649402684945975534001

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