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Properties

Label 2-950-95.4-c1-0-7
Degree $2$
Conductor $950$
Sign $0.405 - 0.914i$
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  + (0.342 − 0.939i)2-s + (−0.402 + 0.0710i)3-s + (−0.766 − 0.642i)4-s + (−0.0710 + 0.402i)6-s + (1.99 + 1.15i)7-s + (−0.866 + 0.500i)8-s + (−2.66 + 0.968i)9-s + (1.32 + 2.29i)11-s + (0.354 + 0.204i)12-s + (−5.02 − 0.885i)13-s + (1.76 − 1.47i)14-s + (0.173 + 0.984i)16-s + (−0.955 + 2.62i)17-s + 2.83i·18-s + (1.11 + 4.21i)19-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.232 + 0.0409i)3-s + (−0.383 − 0.321i)4-s + (−0.0289 + 0.164i)6-s + (0.753 + 0.434i)7-s + (−0.306 + 0.176i)8-s + (−0.887 + 0.322i)9-s + (0.399 + 0.691i)11-s + (0.102 + 0.0590i)12-s + (−1.39 − 0.245i)13-s + (0.471 − 0.395i)14-s + (0.0434 + 0.246i)16-s + (−0.231 + 0.636i)17-s + 0.667i·18-s + (0.256 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.405 - 0.914i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.405 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841751 + 0.547518i\)
\(L(\frac12)\) \(\approx\) \(0.841751 + 0.547518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-1.11 - 4.21i)T \)
good3 \( 1 + (0.402 - 0.0710i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.99 - 1.15i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.02 + 0.885i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.955 - 2.62i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.751 + 0.896i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.18 - 1.52i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.11 - 7.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.99iT - 37T^{2} \)
41 \( 1 + (-1.97 - 11.2i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.990 - 1.18i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.61 + 9.92i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.72 - 3.24i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.5 - 3.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.53 - 2.12i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.88 - 7.91i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.239 - 0.200i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-15.1 + 2.66i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.52 - 8.67i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.53 + 0.887i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.12 - 6.39i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.896 + 2.46i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.28530533053813028417892477854, −9.531011053844172630857413792600, −8.583435417926079011683578339298, −7.83964054134943417791000992812, −6.72926013659236988310750136819, −5.44911772616044729640214640054, −5.06887223929764477524279668103, −3.91273408024330334848709652994, −2.63206566616469645383531928118, −1.68867854916557704634531254008, 0.43272713942311062953523827518, 2.44420466288414000748521929865, 3.71736287482373144401060194148, 4.83517221229608311445220327155, 5.45902524719225576828852511117, 6.49819833713191836949848264124, 7.33084852394013464279632275355, 8.014984982270887045127330638849, 9.091448497549953153309623669019, 9.552869971174819313287944007152

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