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Properties

Label 2-950-95.27-c1-0-22
Degree $2$
Conductor $950$
Sign $-0.520 + 0.853i$
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 + (0.694 − 0.186i)3-s + (−0.866 + 0.499i)4-s + (−0.359 − 0.622i)6-s + (−1.01 − 1.01i)7-s + (0.707 + 0.707i)8-s + (−2.14 + 1.24i)9-s + 2.38·11-s + (−0.508 + 0.508i)12-s + (0.778 − 2.90i)13-s + (−0.717 + 1.24i)14-s + (0.500 − 0.866i)16-s + (4.58 − 1.22i)17-s + (1.75 + 1.75i)18-s + (3.35 − 2.78i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.401 − 0.107i)3-s + (−0.433 + 0.249i)4-s + (−0.146 − 0.254i)6-s + (−0.383 − 0.383i)7-s + (0.249 + 0.249i)8-s + (−0.716 + 0.413i)9-s + 0.720·11-s + (−0.146 + 0.146i)12-s + (0.215 − 0.805i)13-s + (−0.191 + 0.331i)14-s + (0.125 − 0.216i)16-s + (1.11 − 0.298i)17-s + (0.413 + 0.413i)18-s + (0.769 − 0.639i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635021 - 1.13120i\)
\(L(\frac12)\) \(\approx\) \(0.635021 - 1.13120i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.35 + 2.78i)T \)
good3 \( 1 + (-0.694 + 0.186i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.01 + 1.01i)T + 7iT^{2} \)
11 \( 1 - 2.38T + 11T^{2} \)
13 \( 1 + (-0.778 + 2.90i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.58 + 1.22i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (5.07 + 1.35i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.88 + 8.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + (2.87 - 2.87i)T - 37iT^{2} \)
41 \( 1 + (-6.97 - 4.02i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 - 1.76i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0935 - 0.349i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.89 + 7.08i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.29 - 2.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.310 + 0.537i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.37 - 0.368i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.38 + 3.11i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.862 - 3.21i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.0163 + 0.0282i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.0 + 12.0i)T - 83iT^{2} \)
89 \( 1 + (1.11 + 1.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.64 - 9.85i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.728998164065787527037648279769, −9.161467475891599212502062075956, −7.943217020310119536494062124454, −7.70633826304061531974793145930, −6.23170368598862786399818454139, −5.38529807059639903045941193118, −4.05678566333927982666646437279, −3.23763256211313940112930870519, −2.23735856531872416060944762201, −0.64693823263279695743784677760, 1.51984267183018848119659203254, 3.22969484924434120885828697804, 3.93634172274569640839322971466, 5.41961863348298634415507362981, 6.00079024423279451181701900885, 6.96221634243379399983320760430, 7.82064169260951687986034795662, 8.850046361162230265281769842644, 9.168196053472278760117695556459, 10.02030090544074170361709121685

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