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Properties

Label 2-650-65.32-c1-0-3
Degree $2$
Conductor $650$
Sign $-0.500 + 0.865i$
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  + (−0.5 + 0.866i)2-s + (0.876 + 3.27i)3-s + (−0.499 − 0.866i)4-s + (−3.27 − 0.876i)6-s + (−3.61 + 2.08i)7-s + 0.999·8-s + (−7.33 + 4.23i)9-s + (−0.732 + 0.196i)11-s + (2.39 − 2.39i)12-s + (3.59 + 0.333i)13-s − 4.17i·14-s + (−0.5 + 0.866i)16-s + (2.47 + 0.662i)17-s − 8.46i·18-s + (1.24 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.505 + 1.88i)3-s + (−0.249 − 0.433i)4-s + (−1.33 − 0.357i)6-s + (−1.36 + 0.789i)7-s + 0.353·8-s + (−2.44 + 1.41i)9-s + (−0.220 + 0.0591i)11-s + (0.691 − 0.691i)12-s + (0.995 + 0.0924i)13-s − 1.11i·14-s + (−0.125 + 0.216i)16-s + (0.599 + 0.160i)17-s − 1.99i·18-s + (0.286 − 1.07i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.423379 - 0.733746i\)
\(L(\frac12)\) \(\approx\) \(0.423379 - 0.733746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.59 - 0.333i)T \)
good3 \( 1 + (-0.876 - 3.27i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.61 - 2.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.732 - 0.196i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.47 - 0.662i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.24 + 4.66i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.01 - 0.538i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.35 - 2.51i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.70 - 1.70i)T - 31iT^{2} \)
37 \( 1 + (1.55 + 0.895i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.417 - 1.55i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.44 - 9.11i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 4.89iT - 47T^{2} \)
53 \( 1 + (3.87 - 3.87i)T - 53iT^{2} \)
59 \( 1 + (12.6 + 3.38i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.00936 + 0.0162i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 - 9.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.24 - 1.94i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 2.26T + 73T^{2} \)
79 \( 1 - 4.98iT - 79T^{2} \)
83 \( 1 - 2.56iT - 83T^{2} \)
89 \( 1 + (1.73 + 6.47i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.50 - 6.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.69097435077110838290662756554, −10.02093963229037719608769934365, −9.292210356292972338708467329996, −8.873437214867278851135802128828, −7.993212426000721102777529272059, −6.46705435411443742045520714936, −5.66351033767313242356182707804, −4.77588564374347727819847479608, −3.58045972108460270638700545108, −2.82974725373952884791402001797, 0.48121120679471329072528673216, 1.65606775874990892625134832744, 3.02373242375398201316576486104, 3.65234577591582624964707932443, 5.89856376146258185304174270051, 6.54483898937082737505716682486, 7.49893983189155888223252656192, 8.087229637145619624742440740260, 9.011254652404727389088339624910, 9.937845454682083706874772519474

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