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Properties

Label 2-950-95.8-c1-0-15
Degree $2$
Conductor $950$
Sign $0.875 - 0.483i$
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  + (0.965 − 0.258i)2-s + (0.517 + 1.93i)3-s + (0.866 − 0.499i)4-s + (0.999 + 1.73i)6-s + (1.22 − 1.22i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + 3·11-s + (1.41 + 1.41i)12-s + (1.93 + 0.517i)13-s + (0.866 − 1.49i)14-s + (0.500 − 0.866i)16-s + (−0.707 + 0.707i)18-s + (−2.59 − 3.5i)19-s + (2.99 + 1.73i)21-s + (2.89 − 0.776i)22-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.298 + 1.11i)3-s + (0.433 − 0.249i)4-s + (0.408 + 0.707i)6-s + (0.462 − 0.462i)7-s + (0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + 0.904·11-s + (0.408 + 0.408i)12-s + (0.535 + 0.143i)13-s + (0.231 − 0.400i)14-s + (0.125 − 0.216i)16-s + (−0.166 + 0.166i)18-s + (−0.596 − 0.802i)19-s + (0.654 + 0.377i)21-s + (0.617 − 0.165i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88435 + 0.743611i\)
\(L(\frac12)\) \(\approx\) \(2.88435 + 0.743611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
19 \( 1 + (2.59 + 3.5i)T \)
good3 \( 1 + (-0.517 - 1.93i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1.93 - 0.517i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.34 - 5.01i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + 37iT^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.69 - 1.79i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-10.0 - 2.68i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.89 - 0.776i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.517 + 1.93i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.34 - 0.896i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.73 - 3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.72 - 2.07i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.20168978222093528953818426960, −9.357673943242789677041395192359, −8.697624778619120888190721691959, −7.49709878986597158163437155181, −6.57214529028806746585505217002, −5.55060595136362095696237883140, −4.42770234790053404069031098247, −4.06309161637617739374551419781, −3.05928605366914409912635178404, −1.50240048113606666344430045103, 1.42586684958721633721027918829, 2.34027934858648274051206662387, 3.64157976228146057145889837642, 4.67103348116361139051894619245, 5.85588213682383532955771343317, 6.54800827539554774600857873205, 7.26977550704708374123066536654, 8.344933490523939447754541803213, 8.634035618088052257785481454324, 10.07658538321780704120065960782

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