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Properties

Label 2-950-95.12-c1-0-21
Degree $2$
Conductor $950$
Sign $0.972 + 0.234i$
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 + (0.108 − 0.405i)3-s + (0.866 + 0.499i)4-s + (0.209 − 0.363i)6-s + (−2.14 − 2.14i)7-s + (0.707 + 0.707i)8-s + (2.44 + 1.41i)9-s + 2.73·11-s + (0.296 − 0.296i)12-s + (−0.676 + 0.181i)13-s + (−1.51 − 2.63i)14-s + (0.500 + 0.866i)16-s + (1.36 − 5.10i)17-s + (1.99 + 1.99i)18-s + (4.06 + 1.58i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.0627 − 0.234i)3-s + (0.433 + 0.249i)4-s + (0.0857 − 0.148i)6-s + (−0.812 − 0.812i)7-s + (0.249 + 0.249i)8-s + (0.815 + 0.470i)9-s + 0.825·11-s + (0.0857 − 0.0857i)12-s + (−0.187 + 0.0502i)13-s + (−0.406 − 0.703i)14-s + (0.125 + 0.216i)16-s + (0.331 − 1.23i)17-s + (0.470 + 0.470i)18-s + (0.931 + 0.362i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.972 + 0.234i$
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: \((2,\ 950,\ (\ :1/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52463 - 0.300775i\)
\(L(\frac12)\) \(\approx\) \(2.52463 - 0.300775i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.06 - 1.58i)T \)
good3 \( 1 + (-0.108 + 0.405i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.14 + 2.14i)T + 7iT^{2} \)
11 \( 1 - 2.73T + 11T^{2} \)
13 \( 1 + (0.676 - 0.181i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-1.36 + 5.10i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.873 - 3.26i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.95 + 5.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.61iT - 31T^{2} \)
37 \( 1 + (-1.70 + 1.70i)T - 37iT^{2} \)
41 \( 1 + (7.48 - 4.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.8 - 2.91i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.93 + 1.05i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.99 - 2.14i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.27 + 5.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.23 - 3.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.52 - 9.41i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.34 + 4.81i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.33 + 2.50i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.46 + 2.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.7 - 10.7i)T - 83iT^{2} \)
89 \( 1 + (8.62 - 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.33 + 1.69i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.783705137167687617936206159526, −9.545556507831784274029136626003, −7.965082909911417730308015114986, −7.28287796686479349900728274559, −6.72539310126777730990431353859, −5.69966213955556305042114556075, −4.58909153517390033955899730728, −3.80202008280915678312811868612, −2.74732311721580576174592908345, −1.16858894948965702405629109224, 1.41611568814724254448749718024, 2.91175317652180724275542421388, 3.70422202216315594564503764512, 4.67823965820255392936168978343, 5.76398466087940282084509554122, 6.52110844759731582873505679317, 7.24367364314728215931017962366, 8.632486533126215890389293689250, 9.329259309100450049491446030204, 10.09144088397973823160504491315

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