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Properties

Label 2-650-65.9-c1-0-18
Degree $2$
Conductor $650$
Sign $0.300 - 0.953i$
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 + (2.08 − 3.60i)11-s + 3.16i·12-s + (−3.60 + 0.0811i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (−1.00 + 0.581i)17-s + 7i·18-s + (−4.08 − 7.06i)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 + (0.627 − 1.08i)11-s + 0.912i·12-s + (−0.999 + 0.0225i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.244 + 0.140i)17-s + 1.64i·18-s + (−0.936 − 1.62i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78248 + 2.03972i\)
\(L(\frac12)\) \(\approx\) \(2.78248 + 2.03972i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 0.0811i)T \)
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 \( 1 + (-2.08 + 3.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.00 - 0.581i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.08 + 7.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.16 - 2.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.837T + 31T^{2} \)
37 \( 1 + (-5.33 - 3.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.16 - 2.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 4.16iT - 53T^{2} \)
59 \( 1 + (-1.16 - 2.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.74 + 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.94 - 5.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.16 + 7.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.16iT - 73T^{2} \)
79 \( 1 + 5.48T + 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + (-2.66 + 4.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.94 - 5.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.67337168896445786795873914271, −9.657820290754750419821967674167, −8.851596796125257115062739849159, −8.258248587415459261942975666958, −7.37227635940960169067534825622, −6.25440417197875228267276308316, −4.75965677126533595258094502267, −4.26813524316404327031208182045, −3.16140014047470707976778994434, −2.29112996767983213730095804152, 1.78109470574208541352716924350, 2.27014195285683095395520097936, 3.64022138466580482654507500178, 4.42827993768968557813650411967, 5.97855611045952120237107197181, 7.01887545851900497886698793730, 7.71685702375440965793406648524, 8.522275267454920376052408177181, 9.612154375933061487766242643961, 10.04616238659341874181544380207

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