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Properties

Label 2-950-95.88-c1-0-3
Degree $2$
Conductor $950$
Sign $0.430 - 0.902i$
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  + (0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s + (−0.499 + 0.866i)6-s + (−0.707 + 0.707i)8-s + (−1.73 − i)9-s − 3·11-s + (0.707 + 0.707i)12-s + (0.517 + 1.93i)13-s + (0.500 + 0.866i)16-s + (6.69 + 1.79i)17-s + (−1.41 + 1.41i)18-s + (−4.33 − 0.5i)19-s + (−0.776 + 2.89i)22-s + (−3.34 + 0.896i)23-s + (0.866 − 0.500i)24-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (−0.204 + 0.353i)6-s + (−0.249 + 0.249i)8-s + (−0.577 − 0.333i)9-s − 0.904·11-s + (0.204 + 0.204i)12-s + (0.143 + 0.535i)13-s + (0.125 + 0.216i)16-s + (1.62 + 0.434i)17-s + (−0.333 + 0.333i)18-s + (−0.993 − 0.114i)19-s + (−0.165 + 0.617i)22-s + (−0.697 + 0.186i)23-s + (0.176 − 0.102i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456311 + 0.288032i\)
\(L(\frac12)\) \(\approx\) \(0.456311 + 0.288032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (4.33 + 0.5i)T \)
good3 \( 1 + (0.965 + 0.258i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-0.517 - 1.93i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.69 - 1.79i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (3.34 - 0.896i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.46 - 6i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-1.41 - 1.41i)T + 37iT^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.896 - 3.34i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.79 - 6.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.55 + 5.79i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.965 - 0.258i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.03 - 15.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.46 - 6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.22 - 1.22i)T + 83iT^{2} \)
89 \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.258 + 0.965i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.37616792777472372747914367914, −9.553674623002500346828637534552, −8.579334431915652441232970803975, −7.79990506296944469611332532374, −6.56744081134635781960520934453, −5.73119444633423297561875497539, −5.00887543432113159723218995621, −3.78012838070115965937355693710, −2.81482612510406637211855109702, −1.41065843688001512946831795246, 0.25977988138159758952764238809, 2.46947952528069924834714475111, 3.71755599536238791887935735126, 4.89360710001650277663498908168, 5.66102632845442124867217263367, 6.14364544853668466210394521764, 7.53313494495659942680350637230, 7.975727289956310964016093520053, 8.876923177085909791875459759302, 10.11429094613421804554911059862

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