Properties

Label 2-650-5.4-c5-0-35
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 + 9.04i·3-s − 16·4-s + 36.1·6-s + 77.0i·7-s + 64i·8-s + 161.·9-s + 463.·11-s − 144. i·12-s + 169i·13-s + 308.·14-s + 256·16-s + 1.86e3i·17-s − 644. i·18-s + 618.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.580i·3-s − 0.5·4-s + 0.410·6-s + 0.594i·7-s + 0.353i·8-s + 0.663·9-s + 1.15·11-s − 0.290i·12-s + 0.277i·13-s + 0.420·14-s + 0.250·16-s + 1.56i·17-s − 0.469i·18-s + 0.392·19-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

\(L(3)\) \(\approx\) \(2.311854262\)
\(L(\frac12)\) \(\approx\) \(2.311854262\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.04iT - 243T^{2} \)
7 \( 1 - 77.0iT - 1.68e4T^{2} \)
11 \( 1 - 463.T + 1.61e5T^{2} \)
17 \( 1 - 1.86e3iT - 1.41e6T^{2} \)
19 \( 1 - 618.T + 2.47e6T^{2} \)
23 \( 1 - 1.71e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.86e3T + 2.05e7T^{2} \)
31 \( 1 + 545.T + 2.86e7T^{2} \)
37 \( 1 + 1.23e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.77e4T + 1.15e8T^{2} \)
43 \( 1 + 1.26e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.59e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.79e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.22e4T + 7.14e8T^{2} \)
61 \( 1 - 5.54e3T + 8.44e8T^{2} \)
67 \( 1 - 5.96e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.72e4T + 1.80e9T^{2} \)
73 \( 1 + 6.73e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.90e4T + 3.07e9T^{2} \)
83 \( 1 + 1.25e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.36e3T + 5.58e9T^{2} \)
97 \( 1 - 5.80e4iT - 8.58e9T^{2} \)
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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

−10.11336037001943811248353186732, −9.080711470518403085407296672574, −8.671214185374675549537889170108, −7.31713844392224610055770436578, −6.23680823340364545875570562925, −5.20519040235512182770908392387, −4.09192726373308880786002949974, −3.56270454889092994946501765573, −2.06599223186808326222511489279, −1.16729531637881375391571567905, 0.57361469626341049233375630817, 1.37769705312942184495820928695, 3.01725868144286100281246537927, 4.27037542542211975411927471827, 5.03539546285517689201827733460, 6.50764220212980506986409170547, 6.82095483841854668086942861749, 7.69787123293199042388872038979, 8.568118250657419652031211602614, 9.621188136452250106392456576217

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