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Properties

Label 2-950-95.49-c1-0-7
Degree $2$
Conductor $950$
Sign $0.999 + 0.00286i$
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 + (0.410 + 0.236i)3-s + (0.499 + 0.866i)4-s + (−0.236 − 0.410i)6-s + 2.19i·7-s − 0.999i·8-s + (−1.38 − 2.40i)9-s − 4.96·11-s + 0.473i·12-s + (1.97 − 1.14i)13-s + (1.09 − 1.89i)14-s + (−0.5 + 0.866i)16-s + (5.86 + 3.38i)17-s + 2.77i·18-s + (3.12 − 3.03i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.236 + 0.136i)3-s + (0.249 + 0.433i)4-s + (−0.0967 − 0.167i)6-s + 0.828i·7-s − 0.353i·8-s + (−0.462 − 0.801i)9-s − 1.49·11-s + 0.136i·12-s + (0.548 − 0.316i)13-s + (0.292 − 0.507i)14-s + (−0.125 + 0.216i)16-s + (1.42 + 0.821i)17-s + 0.654i·18-s + (0.716 − 0.697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.999 + 0.00286i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.999 + 0.00286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21099 - 0.00173467i\)
\(L(\frac12)\) \(\approx\) \(1.21099 - 0.00173467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 + 3.03i)T \)
good3 \( 1 + (-0.410 - 0.236i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.19iT - 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
13 \( 1 + (-1.97 + 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.86 - 3.38i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.65 + 3.84i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.983 - 1.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 - 7.38iT - 37T^{2} \)
41 \( 1 + (5.70 - 9.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.79 - 5.07i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.62 - 2.09i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.62 - 2.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.72 + 2.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.700 - 0.404i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.20 - 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.63 + 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (1.78 + 3.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.149 + 0.0861i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.970458632193443921883980219048, −9.248968648771058060780643584490, −8.297131712217987782663192493375, −8.010121747532105106850689628989, −6.65234283110150869632654467203, −5.75428669353662188368745902033, −4.80251690582403557277848100335, −3.06777706411199137299932922174, −2.88724499389803043389424149933, −1.01797480147208732508748464324, 0.913790865588669623641307549448, 2.45493181662175115890462265523, 3.54405935328690266078444382031, 5.14931196201900064074277895390, 5.55686837493323764451387969125, 7.02746845292647437103357918436, 7.64426263630006131984114456596, 8.129052629369549662559389370367, 9.164266563905798287359458365566, 10.10357195332668089294038511080

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