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Properties

Label 2-950-95.12-c1-0-12
Degree $2$
Conductor $950$
Sign $0.708 - 0.705i$
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  + (−0.965 − 0.258i)2-s + (−0.313 + 1.17i)3-s + (0.866 + 0.499i)4-s + (0.606 − 1.04i)6-s + (1.88 + 1.88i)7-s + (−0.707 − 0.707i)8-s + (1.32 + 0.765i)9-s + 5.08·11-s + (−0.857 + 0.857i)12-s + (−1.59 + 0.428i)13-s + (−1.33 − 2.30i)14-s + (0.500 + 0.866i)16-s + (2.07 − 7.73i)17-s + (−1.08 − 1.08i)18-s + (1.52 − 4.08i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.181 + 0.676i)3-s + (0.433 + 0.249i)4-s + (0.247 − 0.428i)6-s + (0.712 + 0.712i)7-s + (−0.249 − 0.249i)8-s + (0.441 + 0.255i)9-s + 1.53·11-s + (−0.247 + 0.247i)12-s + (−0.443 + 0.118i)13-s + (−0.356 − 0.617i)14-s + (0.125 + 0.216i)16-s + (0.502 − 1.87i)17-s + (−0.255 − 0.255i)18-s + (0.349 − 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25902 + 0.519816i\)
\(L(\frac12)\) \(\approx\) \(1.25902 + 0.519816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (-1.52 + 4.08i)T \)
good3 \( 1 + (0.313 - 1.17i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.88 - 1.88i)T + 7iT^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 + (1.59 - 0.428i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.07 + 7.73i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.16 - 8.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.858 - 1.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.07iT - 31T^{2} \)
37 \( 1 + (0.618 - 0.618i)T - 37iT^{2} \)
41 \( 1 + (-4.93 + 2.84i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.23 + 1.13i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-9.97 + 2.67i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.22 - 1.66i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.18 - 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 - 4.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.363 + 1.35i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (13.2 - 7.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.99 - 0.535i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.01 - 3.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.12 + 3.12i)T - 83iT^{2} \)
89 \( 1 + (5.26 - 9.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 - 2.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.873726005954803361792264571850, −9.317769953181340091424798491093, −8.917167742993473276246978498170, −7.46463869924769068423586200209, −7.15178309775200888897497761119, −5.66374460115640872850655244029, −4.90289735951209858405574032273, −3.86118172864760293950560222860, −2.56329858753916510336434565006, −1.24665773658309937120010863904, 1.07957521550874345604301562930, 1.75630069284740427954888374050, 3.64540533192881503975355917823, 4.56570391261455809826717227698, 6.07812122980397180545655270180, 6.59050829596396293933019258496, 7.47959466720446885429210481963, 8.148330043638553896861419267931, 8.984443400653970955091872507798, 10.02643866920438341130142808163

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