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Properties

Label 2-650-65.7-c1-0-20
Degree $2$
Conductor $650$
Sign $-0.838 + 0.545i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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  + (0.866 + 0.5i)2-s + (−0.814 − 3.04i)3-s + (0.499 + 0.866i)4-s + (0.814 − 3.04i)6-s + (−0.402 − 0.696i)7-s + 0.999i·8-s + (−5.98 + 3.45i)9-s + (−0.778 − 2.90i)11-s + (2.22 − 2.22i)12-s + (−0.206 − 3.59i)13-s − 0.804i·14-s + (−0.5 + 0.866i)16-s + (−6.99 − 1.87i)17-s − 6.91·18-s + (4.51 + 1.21i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.470 − 1.75i)3-s + (0.249 + 0.433i)4-s + (0.332 − 1.24i)6-s + (−0.152 − 0.263i)7-s + 0.353i·8-s + (−1.99 + 1.15i)9-s + (−0.234 − 0.876i)11-s + (0.642 − 0.642i)12-s + (−0.0571 − 0.998i)13-s − 0.215i·14-s + (−0.125 + 0.216i)16-s + (−1.69 − 0.454i)17-s − 1.62·18-s + (1.03 + 0.277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.838 + 0.545i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345517 - 1.16510i\)
\(L(\frac12)\) \(\approx\) \(0.345517 - 1.16510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (0.206 + 3.59i)T \)
good3 \( 1 + (0.814 + 3.04i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.402 + 0.696i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.778 + 2.90i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (6.99 + 1.87i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.51 - 1.21i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.422 - 0.113i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.58 + 2.06i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.536 + 0.536i)T + 31iT^{2} \)
37 \( 1 + (-0.482 + 0.835i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.63 + 0.437i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.64 - 6.14i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 + (-5.01 + 5.01i)T - 53iT^{2} \)
59 \( 1 + (0.0422 - 0.157i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.11 - 1.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.82 + 2.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.63 + 9.83i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 7.75iT - 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (-5.91 + 1.58i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-11.6 + 6.70i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58157663392177044855428321182, −8.998853714987624038073981178599, −7.982320701919979270819050646007, −7.45946042383996819405579923029, −6.54818543143324435628073737378, −5.88384174909396429511173744695, −5.02370521219296687734691477710, −3.32434496101751503097473184366, −2.20172376459937601146581244803, −0.53962313856119950807947647041, 2.33081727698066044745539208893, 3.66756514344620236651592901407, 4.45144524806738650561352047889, 5.08909452860551641586869615442, 6.08044937328381527938327801793, 7.10447867783756659875793753184, 8.844883170743030162909187271150, 9.369447505663176493549214433048, 10.15126920760642983905437499793, 10.94389383006248917727731074690

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