Properties

Label 2-650-5.4-c5-0-79
Degree $2$
Conductor $650$
Sign $-0.447 + 0.894i$
Analytic cond. $104.249$
Root an. cond. $10.2102$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 3.22i·3-s − 16·4-s + 12.8·6-s − 200. i·7-s − 64i·8-s + 232.·9-s − 530.·11-s + 51.5i·12-s + 169i·13-s + 803.·14-s + 256·16-s − 15.1i·17-s + 930. i·18-s + 392.·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.206i·3-s − 0.5·4-s + 0.146·6-s − 1.54i·7-s − 0.353i·8-s + 0.957·9-s − 1.32·11-s + 0.103i·12-s + 0.277i·13-s + 1.09·14-s + 0.250·16-s − 0.0126i·17-s + 0.676i·18-s + 0.249·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(104.249\)
Root analytic conductor: \(10.2102\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.175639959\)
\(L(\frac12)\) \(\approx\) \(1.175639959\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 \)
13 \( 1 - 169iT \)
good3 \( 1 + 3.22iT - 243T^{2} \)
7 \( 1 + 200. iT - 1.68e4T^{2} \)
11 \( 1 + 530.T + 1.61e5T^{2} \)
17 \( 1 + 15.1iT - 1.41e6T^{2} \)
19 \( 1 - 392.T + 2.47e6T^{2} \)
23 \( 1 + 2.63e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.13e3T + 2.05e7T^{2} \)
31 \( 1 - 6.82e3T + 2.86e7T^{2} \)
37 \( 1 + 1.32e4iT - 6.93e7T^{2} \)
41 \( 1 - 3.21e3T + 1.15e8T^{2} \)
43 \( 1 - 1.10e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.25e3iT - 2.29e8T^{2} \)
53 \( 1 + 3.52e3iT - 4.18e8T^{2} \)
59 \( 1 + 4.03e4T + 7.14e8T^{2} \)
61 \( 1 + 4.42e4T + 8.44e8T^{2} \)
67 \( 1 - 7.07e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.62e4T + 1.80e9T^{2} \)
73 \( 1 - 4.10e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.94e4T + 3.07e9T^{2} \)
83 \( 1 + 6.75e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.33e4T + 5.58e9T^{2} \)
97 \( 1 + 2.20e3iT - 8.58e9T^{2} \)
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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

−9.597316264009758334032716786219, −8.277769652208717518294035809061, −7.61498839742408040073153062504, −6.98757781218594973133115870854, −6.13172444207821644923400486541, −4.68940879937535060625406298549, −4.30542046913951442647374910370, −2.85084614691639572370735438477, −1.22022309103391759501140567243, −0.27551089067402655469312379277, 1.28978000147736598621030073539, 2.47354201959764268132148851212, 3.17746977429690549705256963460, 4.66197663612106823366654314072, 5.27502655931792704818662700044, 6.32225895189913788447326419851, 7.71027238727486442320775011105, 8.440191531470252838850882259978, 9.382865357404957939439336053971, 10.10941686365408262597627426219

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