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Properties

Label 2-950-95.27-c1-0-28
Degree $2$
Conductor $950$
Sign $-0.706 + 0.707i$
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 + (3.03 − 0.814i)3-s + (−0.866 + 0.499i)4-s + (−1.57 − 2.72i)6-s + (−2.05 − 2.05i)7-s + (0.707 + 0.707i)8-s + (5.96 − 3.44i)9-s − 5.34·11-s + (−2.22 + 2.22i)12-s + (1.00 − 3.75i)13-s + (−1.45 + 2.51i)14-s + (0.500 − 0.866i)16-s + (2.56 − 0.687i)17-s + (−4.87 − 4.87i)18-s + (−3.18 − 2.97i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (1.75 − 0.469i)3-s + (−0.433 + 0.249i)4-s + (−0.641 − 1.11i)6-s + (−0.777 − 0.777i)7-s + (0.249 + 0.249i)8-s + (1.98 − 1.14i)9-s − 1.61·11-s + (−0.641 + 0.641i)12-s + (0.279 − 1.04i)13-s + (−0.388 + 0.673i)14-s + (0.125 − 0.216i)16-s + (0.622 − 0.166i)17-s + (−1.14 − 1.14i)18-s + (−0.730 − 0.682i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.706 + 0.707i$
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: \((2,\ 950,\ (\ :1/2),\ -0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791857 - 1.90893i\)
\(L(\frac12)\) \(\approx\) \(0.791857 - 1.90893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.18 + 2.97i)T \)
good3 \( 1 + (-3.03 + 0.814i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.05 + 2.05i)T + 7iT^{2} \)
11 \( 1 + 5.34T + 11T^{2} \)
13 \( 1 + (-1.00 + 3.75i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.56 + 0.687i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-2.80 - 0.752i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.53 - 6.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.91iT - 31T^{2} \)
37 \( 1 + (3.67 - 3.67i)T - 37iT^{2} \)
41 \( 1 + (-4.61 - 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.74 - 6.51i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.622 + 2.32i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.917 + 3.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.58 + 7.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.199 - 0.345i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0355 - 0.00951i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.91 - 2.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.16 - 15.5i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.15 + 1.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.45 + 3.45i)T - 83iT^{2} \)
89 \( 1 + (-6.68 - 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.59 - 13.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.830088341046468986067352766432, −8.867338012690981962640587302673, −8.055791161353791884957602381909, −7.62306061343080290171423555875, −6.66087954303855204475756326185, −5.09658662962307474094993605717, −3.80371402648376839451016200312, −3.03345996468168181883033302677, −2.43963723788457692717770721810, −0.828074666296963474884691264127, 2.11654436493519132353888765115, 3.00187145986531243043824249515, 4.00183948721249909173362701913, 5.06480493096350199102006258243, 6.16301886768548969306818075133, 7.27378634236939640447011987342, 7.996315854973383227470159635436, 8.779741321836720028041901604720, 9.143630351798926660308173129975, 10.14956799526280951181058194346

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