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Properties

Label 2-650-13.10-c1-0-9
Degree $2$
Conductor $650$
Sign $0.996 - 0.0791i$
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 + (0.547 + 0.948i)3-s + (0.499 − 0.866i)4-s + (0.948 + 0.547i)6-s + (3.45 + 1.99i)7-s − 0.999i·8-s + (0.900 − 1.56i)9-s + (0.142 − 0.0820i)11-s + 1.09·12-s + (−1.50 + 3.27i)13-s + 3.99·14-s + (−0.5 − 0.866i)16-s + (−0.783 + 1.35i)17-s − 1.80i·18-s + (−0.716 − 0.413i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.316 + 0.547i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)6-s + (1.30 + 0.754i)7-s − 0.353i·8-s + (0.300 − 0.520i)9-s + (0.0428 − 0.0247i)11-s + 0.316·12-s + (−0.418 + 0.908i)13-s + 1.06·14-s + (−0.125 − 0.216i)16-s + (−0.190 + 0.329i)17-s − 0.424i·18-s + (−0.164 − 0.0948i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.996 - 0.0791i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.996 - 0.0791i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63896 + 0.104558i\)
\(L(\frac12)\) \(\approx\) \(2.63896 + 0.104558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 3.27i)T \)
good3 \( 1 + (-0.547 - 0.948i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.45 - 1.99i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.142 + 0.0820i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.783 - 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.716 + 0.413i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.49 + 6.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.92 - 5.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.0858iT - 31T^{2} \)
37 \( 1 + (-7.24 + 4.18i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.48 + 4.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.62 - 9.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.31iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.21 + 5.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.2 - 7.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.179iT - 73T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + 1.23iT - 83T^{2} \)
89 \( 1 + (8.98 - 5.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.3 + 9.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71407706528054353108211194647, −9.718498009462341264258942079576, −8.925211859849248036057885388190, −8.140074254829724453837690301641, −6.85377284327656809328783563690, −5.88737179559450044289677808889, −4.64604895377209324792638491463, −4.28375870011145302700452508357, −2.81181800927902652743995001013, −1.69123191845941668783094270131, 1.47663793573767851387330753531, 2.73613077638979320478369394227, 4.22340527173897306680181634371, 4.92249436306454599685736199559, 5.99681189718748457297686529319, 7.26646980183998176268650277572, 7.77770736054427764970417503341, 8.280369931050408686174302303089, 9.761460630119848197832504219144, 10.69535158434322352096512765680

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