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Properties

Label 2-650-25.21-c1-0-24
Degree $2$
Conductor $650$
Sign $0.0897 + 0.995i$
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  + (0.309 − 0.951i)2-s + (0.859 + 0.624i)3-s + (−0.809 − 0.587i)4-s + (1.70 − 1.44i)5-s + (0.859 − 0.624i)6-s + 0.905·7-s + (−0.809 + 0.587i)8-s + (−0.578 − 1.78i)9-s + (−0.851 − 2.06i)10-s + (1.12 − 3.46i)11-s + (−0.328 − 1.00i)12-s + (0.309 + 0.951i)13-s + (0.279 − 0.861i)14-s + (2.36 − 0.181i)15-s + (0.309 + 0.951i)16-s + (−2.10 + 1.53i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.496 + 0.360i)3-s + (−0.404 − 0.293i)4-s + (0.761 − 0.647i)5-s + (0.350 − 0.254i)6-s + 0.342·7-s + (−0.286 + 0.207i)8-s + (−0.192 − 0.593i)9-s + (−0.269 − 0.653i)10-s + (0.339 − 1.04i)11-s + (−0.0947 − 0.291i)12-s + (0.0857 + 0.263i)13-s + (0.0747 − 0.230i)14-s + (0.611 − 0.0467i)15-s + (0.0772 + 0.237i)16-s + (−0.510 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.0897 + 0.995i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.0897 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57020 - 1.43512i\)
\(L(\frac12)\) \(\approx\) \(1.57020 - 1.43512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-1.70 + 1.44i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.859 - 0.624i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.905T + 7T^{2} \)
11 \( 1 + (-1.12 + 3.46i)T + (-8.89 - 6.46i)T^{2} \)
17 \( 1 + (2.10 - 1.53i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.976 - 0.709i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.02 - 3.14i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.76 - 2.00i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.05 + 4.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.189 + 0.583i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.459 - 1.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + (-3.99 - 2.90i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.47 - 5.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0784 - 0.241i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.479 - 1.47i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.20 + 4.50i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.05 - 0.769i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.54 - 10.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.31 + 0.957i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.44 - 1.04i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.30 - 7.10i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.299 + 0.217i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.23728082604365718272028897569, −9.499258745133838016555463454247, −8.765239874873835228323742246275, −8.232112360842790331463080873987, −6.46073939951970579842685485260, −5.73799881296680046285905422393, −4.57060945370600263880924538666, −3.67788023045087715099977855658, −2.49623339245453194683850394865, −1.12502101202451070109099829755, 1.93232137023428591461266714640, 2.91313947838560275157800571118, 4.46527469625911185904603376189, 5.33222275183013283229334439442, 6.54693019340790673285520668024, 7.06740198414735701613604607115, 8.070287278565046684146760196074, 8.806928553631085021092664366084, 9.857854840986287951578164835566, 10.56930657460890405067698109431

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