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Properties

Label 2-950-25.21-c1-0-36
Degree $2$
Conductor $950$
Sign $0.929 + 0.368i$
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  + (−0.309 + 0.951i)2-s + (1 + 0.726i)3-s + (−0.809 − 0.587i)4-s + (1.80 − 1.31i)5-s + (−1 + 0.726i)6-s − 0.763·7-s + (0.809 − 0.587i)8-s + (−0.454 − 1.40i)9-s + (0.690 + 2.12i)10-s + (0.618 − 1.90i)11-s + (−0.381 − 1.17i)12-s + (−1.5 − 4.61i)13-s + (0.236 − 0.726i)14-s + 2.76·15-s + (0.309 + 0.951i)16-s + (−0.5 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.577 + 0.419i)3-s + (−0.404 − 0.293i)4-s + (0.809 − 0.587i)5-s + (−0.408 + 0.296i)6-s − 0.288·7-s + (0.286 − 0.207i)8-s + (−0.151 − 0.466i)9-s + (0.218 + 0.672i)10-s + (0.186 − 0.573i)11-s + (−0.110 − 0.339i)12-s + (−0.416 − 1.28i)13-s + (0.0630 − 0.194i)14-s + 0.713·15-s + (0.0772 + 0.237i)16-s + (−0.121 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.929 + 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61845 - 0.308736i\)
\(L(\frac12)\) \(\approx\) \(1.61845 - 0.308736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-1 - 0.726i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.763T + 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-0.145 + 0.449i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.73 + 3.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.618 + 0.449i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.04 - 3.21i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.881 - 2.71i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (-5.61 - 4.08i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.92 - 2.85i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.854 + 2.62i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.11 + 9.59i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.61 - 1.17i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.23 - 6.71i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.73 - 11.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.23 + 5.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.47 + 3.97i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.33 - 7.19i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.35 - 3.88i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.738204789272172895812859858557, −9.148830367061403060974623155477, −8.431395025712084432783119559290, −7.66759940992418323650880989779, −6.38444687630726573561616966896, −5.79484136771367539894364965171, −4.86728394238830128149742754275, −3.69169540482573728753052479780, −2.59002684954284807726476175964, −0.77836740431549742667461865387, 1.76059467995092779791007028010, 2.34252574555357187049525370399, 3.43788491443294279198805675644, 4.66346085841131117763941521654, 5.78965948566742350599513996242, 7.01133261879057074846407707745, 7.41940569439564237787800967935, 8.732968738910852680660796582946, 9.300276004417372517402880733003, 10.01686500130278025701304425379

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