Properties

Label 2-650-65.4-c1-0-16
Degree $2$
Conductor $650$
Sign $-0.743 - 0.668i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−2.36 − 1.36i)3-s + (−0.499 − 0.866i)4-s + (2.36 − 1.36i)6-s + (−1.5 − 2.59i)7-s + 0.999·8-s + (2.23 + 3.86i)9-s + (−2.59 − 1.5i)11-s + 2.73i·12-s + (−0.866 − 3.5i)13-s + 3·14-s + (−0.5 + 0.866i)16-s + (−1.90 + 1.09i)17-s − 4.46·18-s + (5.59 − 3.23i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−1.36 − 0.788i)3-s + (−0.249 − 0.433i)4-s + (0.965 − 0.557i)6-s + (−0.566 − 0.981i)7-s + 0.353·8-s + (0.744 + 1.28i)9-s + (−0.783 − 0.452i)11-s + 0.788i·12-s + (−0.240 − 0.970i)13-s + 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.461 + 0.266i)17-s − 1.05·18-s + (1.28 − 0.741i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (0.866 + 3.5i)T \)
good3 \( 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.90 - 1.09i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.59 + 3.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.19 - 1.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.73 - 8.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (5.59 - 9.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9 + 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 6.46iT - 53T^{2} \)
59 \( 1 + (9 - 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.09 - 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.66T + 73T^{2} \)
79 \( 1 + 6.19T + 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 + (-14.8 - 8.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.56 - 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38032062665281410067232653636, −9.122879752938344028885819740852, −7.87802713893249143870431104235, −7.16438476931821173747044639742, −6.63956736882643416873447403263, −5.49913138226375341334567695097, −5.03825903277924932348094937811, −3.27300929266387735162797712751, −1.14405864149249001658491644676, 0, 2.16142265383114245846320435162, 3.58528763319669758582063322882, 4.78421049795672838659348076040, 5.47620367881593324017272976935, 6.44919578472906216960667008208, 7.56184184617695768739686710516, 8.920832176278078625597574978559, 9.663237151803568650726055637205, 10.15157804758433485432169275096

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