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Properties

Label 2-650-65.29-c1-0-5
Degree $2$
Conductor $650$
Sign $0.641 - 0.767i$
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 + (−2.73 + 1.58i)3-s + (0.499 − 0.866i)4-s + (−1.58 + 2.73i)6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (3.5 − 6.06i)9-s + (−1.08 − 1.87i)11-s + 3.16i·12-s + (1.87 + 3.08i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (4.47 + 2.58i)17-s − 7i·18-s + (−0.918 + 1.59i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−1.58 + 0.912i)3-s + (0.249 − 0.433i)4-s + (−0.645 + 1.11i)6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (1.16 − 2.02i)9-s + (−0.325 − 0.564i)11-s + 0.912i·12-s + (0.519 + 0.854i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (1.08 + 0.626i)17-s − 1.64i·18-s + (−0.210 + 0.365i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.641 - 0.767i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.641 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13291 + 0.529392i\)
\(L(\frac12)\) \(\approx\) \(1.13291 + 0.529392i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 - 3.08i)T \)
good3 \( 1 + (2.73 - 1.58i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.47 - 2.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.918 - 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.16 - 8.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.16T + 31T^{2} \)
37 \( 1 + (0.140 - 0.0811i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.16 - 8.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.73 - i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 + (5.16 - 8.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.01 - 1.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.16 + 3.74i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.83iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + (3.66 + 6.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.48 - 3.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.66987319156044382163303531677, −10.28164855236339644311092140009, −9.311813709518137652366733934672, −8.063095418297448619790562359160, −6.53164740224662755121743837483, −5.97026846210773328047820542689, −5.15448603842190526671294501174, −4.32027691673740982822949047118, −3.38276409419279589549880129377, −1.30261096195638406398819862053, 0.791001153439965911868632167219, 2.43970486719644768403588950239, 4.29066465557724661962237668359, 5.15592453985918369874978635179, 5.94620176547499199540991523362, 6.61009562929756422181368607054, 7.66411334823391476166721047118, 8.073796187888732037452517634279, 9.960155496616635274052755189129, 10.66210104687976548779825390078

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