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Properties

Label 2-950-95.88-c1-0-2
Degree $2$
Conductor $950$
Sign $0.309 - 0.950i$
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 + (−0.770 − 0.206i)3-s + (−0.866 − 0.499i)4-s + (−0.398 + 0.690i)6-s + (0.349 − 0.349i)7-s + (−0.707 + 0.707i)8-s + (−2.04 − 1.18i)9-s − 1.21·11-s + (0.563 + 0.563i)12-s + (1.75 + 6.55i)13-s + (−0.246 − 0.427i)14-s + (0.500 + 0.866i)16-s + (−5.86 − 1.57i)17-s + (−1.67 + 1.67i)18-s + (4.09 + 1.48i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.444 − 0.119i)3-s + (−0.433 − 0.249i)4-s + (−0.162 + 0.281i)6-s + (0.131 − 0.131i)7-s + (−0.249 + 0.249i)8-s + (−0.682 − 0.394i)9-s − 0.367·11-s + (0.162 + 0.162i)12-s + (0.487 + 1.81i)13-s + (−0.0659 − 0.114i)14-s + (0.125 + 0.216i)16-s + (−1.42 − 0.381i)17-s + (−0.394 + 0.394i)18-s + (0.940 + 0.340i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.309 - 0.950i$
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: \((2,\ 950,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446582 + 0.324289i\)
\(L(\frac12)\) \(\approx\) \(0.446582 + 0.324289i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.09 - 1.48i)T \)
good3 \( 1 + (0.770 + 0.206i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.349 + 0.349i)T - 7iT^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + (-1.75 - 6.55i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (5.86 + 1.57i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (8.86 - 2.37i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.16 + 5.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.84iT - 31T^{2} \)
37 \( 1 + (-2.41 - 2.41i)T + 37iT^{2} \)
41 \( 1 + (3.55 - 2.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.50 - 5.62i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.0539 - 0.201i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.51 - 13.1i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.0144 - 0.0250i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.90 + 6.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.3 - 3.56i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.99 - 3.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.01 + 7.53i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.17 + 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.36 - 2.36i)T + 83iT^{2} \)
89 \( 1 + (-2.10 + 3.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.821 - 3.06i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.31955340712180654158149501388, −9.408308686017116394814954289283, −8.792875641318215070591024416938, −7.75031929986526560212221355357, −6.52649595405954303298742349498, −5.96992387921269407532548165823, −4.73662351998753325287074359331, −4.00656906887473711547990963813, −2.72764788524431518479513734638, −1.52026915815209234198896534114, 0.25360301808575580223198597925, 2.44934078808141056046649919289, 3.65727817037252741213366010591, 4.89203193374795542313254620086, 5.57834001842856176781432582801, 6.22744191199126393540549150209, 7.34086701547496525562808742013, 8.314029950148098198611371418899, 8.613111301496104838843955375790, 10.03468013347057409521133345927

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