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Properties

Label 2-950-95.88-c1-0-5
Degree $2$
Conductor $950$
Sign $-0.566 - 0.823i$
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 + (−1.93 − 0.517i)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 + (−0.517 − 1.93i)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 + (−0.776 + 2.89i)22-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−1.11 − 0.298i)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 + (−0.143 − 0.535i)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 + (−0.165 + 0.617i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.566 - 0.823i$
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: \((2,\ 950,\ (\ :1/2),\ -0.566 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251078 + 0.477442i\)
\(L(\frac12)\) \(\approx\) \(0.251078 + 0.477442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 3.5i)T \)
good3 \( 1 + (1.93 + 0.517i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.22 - 1.22i)T - 7iT^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (0.517 + 1.93i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (5.01 - 1.34i)T + (19.9 - 11.5i)T^{2} \)
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 \( 1 + (1.79 - 6.69i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.68 - 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.776 + 2.89i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (1.93 - 0.517i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.896 - 3.34i)T + (-63.2 - 36.5i)T^{2} \)
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 \( 1 + (-2.07 + 7.72i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.23776778737929690982867571832, −9.437284472740560154512520371698, −8.678393200212373830111268053904, −7.60359437503222920707464121234, −6.73409783094044537979417146234, −6.09465789954059084494611074074, −5.44786511862809973662864899462, −4.43980468799998130953209957656, −3.03724151917399533094724814541, −1.14503804391482223726134189553, 0.36610345342660147689362679311, 1.89881302387080737577765373873, 3.54069216300378697859445836982, 4.26235062956130890248116802382, 5.34523390134475978702146790412, 6.23232145452876893455639826795, 7.06357244449403390724493284919, 8.217873161808356872044736223989, 9.210985559949757195928009947081, 10.05833785169710241358396434179

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