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Properties

Label 2-650-13.5-c2-0-7
Degree $2$
Conductor $650$
Sign $0.471 - 0.881i$
Analytic cond. $17.7112$
Root an. cond. $4.20846$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 0.449·3-s − 2i·4-s + (0.449 − 0.449i)6-s + (−6.67 − 6.67i)7-s + (2 + 2i)8-s − 8.79·9-s + (−10.0 − 10.0i)11-s + 0.898i·12-s + 13i·13-s + 13.3·14-s − 4·16-s + 11.6i·17-s + (8.79 − 8.79i)18-s + (6 − 6i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.149·3-s − 0.5i·4-s + (0.0749 − 0.0749i)6-s + (−0.953 − 0.953i)7-s + (0.250 + 0.250i)8-s − 0.977·9-s + (−0.911 − 0.911i)11-s + 0.0749i·12-s + i·13-s + 0.953·14-s − 0.250·16-s + 0.688i·17-s + (0.488 − 0.488i)18-s + (0.315 − 0.315i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(17.7112\)
Root analytic conductor: \(4.20846\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7287641756\)
\(L(\frac12)\) \(\approx\) \(0.7287641756\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 \)
13 \( 1 - 13iT \)
good3 \( 1 + 0.449T + 9T^{2} \)
7 \( 1 + (6.67 + 6.67i)T + 49iT^{2} \)
11 \( 1 + (10.0 + 10.0i)T + 121iT^{2} \)
17 \( 1 - 11.6iT - 289T^{2} \)
19 \( 1 + (-6 + 6i)T - 361iT^{2} \)
23 \( 1 - 33.3iT - 529T^{2} \)
29 \( 1 - 48.3T + 841T^{2} \)
31 \( 1 + (-11.3 + 11.3i)T - 961iT^{2} \)
37 \( 1 + (-31.6 - 31.6i)T + 1.36e3iT^{2} \)
41 \( 1 + (-45.3 + 45.3i)T - 1.68e3iT^{2} \)
43 \( 1 - 15.9iT - 1.84e3T^{2} \)
47 \( 1 + (59.4 + 59.4i)T + 2.20e3iT^{2} \)
53 \( 1 - 79.6T + 2.80e3T^{2} \)
59 \( 1 + (-53.4 - 53.4i)T + 3.48e3iT^{2} \)
61 \( 1 + 69.0T + 3.72e3T^{2} \)
67 \( 1 + (-58.7 + 58.7i)T - 4.48e3iT^{2} \)
71 \( 1 + (44.7 - 44.7i)T - 5.04e3iT^{2} \)
73 \( 1 + (-67.7 - 67.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 74.7T + 6.24e3T^{2} \)
83 \( 1 + (5.41 - 5.41i)T - 6.88e3iT^{2} \)
89 \( 1 + (-35.6 - 35.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (61.3 - 61.3i)T - 9.40e3iT^{2} \)
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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

−10.37529486794839543027611417786, −9.637556491816740013763429270536, −8.677991461213813686732410309318, −7.926677962642904612056574459843, −6.89958857788135326461171161697, −6.17098340961632319326297954163, −5.28317085063561384055071703613, −3.89676859123201423275823220495, −2.74721111229795710243519719486, −0.806427281464001714615918895965, 0.46187142096878385648108729951, 2.66134003832203711050747751742, 2.86093280876958647625930752539, 4.68231494132880537791801483997, 5.69393507973294220170367814025, 6.60051061002178159112726842337, 7.84581300320328870073712281091, 8.520130736894936420807175375979, 9.478461870299103958398738490557, 10.13866929839026454870322273246

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