Properties

Label 2-650-5.4-c1-0-16
Degree $2$
Conductor $650$
Sign $-0.894 - 0.447i$
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  i·2-s − 2i·3-s − 4-s − 2·6-s − 5i·7-s + i·8-s − 9-s − 3·11-s + 2i·12-s + i·13-s − 5·14-s + 16-s − 3i·17-s + i·18-s + 4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 1.88i·7-s + 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.577i·12-s + 0.277i·13-s − 1.33·14-s + 0.250·16-s − 0.727i·17-s + 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262689 + 1.11276i\)
\(L(\frac12)\) \(\approx\) \(0.262689 + 1.11276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
13 \( 1 - iT \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + 5iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 15iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.08539463281136176625728688899, −9.493397950989875749617801935385, −7.919052387387748458797192749329, −7.52612911527824084085126204832, −6.80236211466351423378386627784, −5.39995479491548125156213249436, −4.24721484283981736705875816276, −3.17247211195596924163512417828, −1.71702188709885502508179621959, −0.63195364148332238163188775522, 2.44979204919754918989041777826, 3.65565428087899144511564961449, 4.98463963356977790824644265893, 5.42171551094545322639681337571, 6.33748866561646631587654187170, 7.73970694839697808678448658724, 8.584494492550617479988325055388, 9.218207064177277851178432758955, 10.01381352146786294455727040879, 10.80263401229139693958530363711

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