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Properties

Label 2-950-19.7-c1-0-13
Degree $2$
Conductor $950$
Sign $0.823 - 0.567i$
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  + (−0.5 − 0.866i)2-s + (1.17 + 2.02i)3-s + (−0.499 + 0.866i)4-s + (1.17 − 2.02i)6-s + 1.34·7-s + 0.999·8-s + (−1.23 + 2.14i)9-s + 3.25·11-s − 2.34·12-s + (0.745 − 1.29i)13-s + (−0.670 − 1.16i)14-s + (−0.5 − 0.866i)16-s + (3.29 + 5.70i)17-s + 2.47·18-s + (−1.25 − 4.17i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.675 + 1.17i)3-s + (−0.249 + 0.433i)4-s + (0.477 − 0.827i)6-s + 0.506·7-s + 0.353·8-s + (−0.413 + 0.715i)9-s + 0.982·11-s − 0.675·12-s + (0.206 − 0.358i)13-s + (−0.179 − 0.310i)14-s + (−0.125 − 0.216i)16-s + (0.798 + 1.38i)17-s + 0.584·18-s + (−0.288 − 0.957i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76486 + 0.549385i\)
\(L(\frac12)\) \(\approx\) \(1.76486 + 0.549385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
19 \( 1 + (1.25 + 4.17i)T \)
good3 \( 1 + (-1.17 - 2.02i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.34T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + (-0.745 + 1.29i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.29 - 5.70i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 - 5.27T + 37T^{2} \)
41 \( 1 + (2.79 + 4.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.81 - 6.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.465 - 0.806i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.54 + 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0837 - 0.145i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.84 - 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.53 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.59 + 6.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.62 - 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.75 - 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.313T + 83T^{2} \)
89 \( 1 + (-4.90 + 8.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.01 + 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.09610162225953416350919281447, −9.303819394049816905325241597825, −8.704188576004277091753792833374, −8.057463976542980090470090119998, −6.85125701501058059134628919873, −5.55173358525728416565860043670, −4.40499983424498971695361331160, −3.80065767315319282519855106919, −2.85414244334450816697052203412, −1.41826683626808175507420826932, 1.09995846833724528937780258188, 2.07436329340933401727746253181, 3.51903058048760560022592606425, 4.79886233939584384982699389994, 5.98309372086324687099309676906, 6.73927087508910683646528700633, 7.67919720691043270948817202291, 7.917845938528794579926917369401, 9.067026008459317522960140901795, 9.501588699813466176123547767850

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