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Properties

Label 2-650-13.3-c1-0-6
Degree $2$
Conductor $650$
Sign $0.252 - 0.967i$
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 + (1.15 + 1.99i)3-s + (−0.499 + 0.866i)4-s + (1.15 − 1.99i)6-s + (−0.348 + 0.603i)7-s + 0.999·8-s + (−1.15 + 1.99i)9-s + (0.348 + 0.603i)11-s − 2.30·12-s + (1.80 + 3.12i)13-s + 0.697·14-s + (−0.5 − 0.866i)16-s + (−1.45 + 2.51i)17-s + 2.30·18-s + (0.197 − 0.341i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.664 + 1.15i)3-s + (−0.249 + 0.433i)4-s + (0.470 − 0.814i)6-s + (−0.131 + 0.228i)7-s + 0.353·8-s + (−0.383 + 0.664i)9-s + (0.105 + 0.182i)11-s − 0.664·12-s + (0.499 + 0.866i)13-s + 0.186·14-s + (−0.125 − 0.216i)16-s + (−0.352 + 0.610i)17-s + 0.542·18-s + (0.0452 − 0.0783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11464 + 0.860989i\)
\(L(\frac12)\) \(\approx\) \(1.11464 + 0.860989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 3.12i)T \)
good3 \( 1 + (-1.15 - 1.99i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.348 - 0.603i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.348 - 0.603i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.197 + 0.341i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.65 + 2.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.60T + 31T^{2} \)
37 \( 1 + (-1.84 - 3.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.25 - 7.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (-5.10 + 8.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.10 - 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.80 + 4.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.30 + 9.18i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 9.30T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + (0.0458 + 0.0793i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.84 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.73454097746669170711458570143, −9.584922044402919137238433419718, −9.309482638301263876540399676179, −8.539537636595280823790047592797, −7.48825247924549303944075070225, −6.21713960303585781515923557192, −4.85169619639647283815336135575, −3.95129357185120341428233237020, −3.19258275711426987535472044173, −1.81804281601922688460396764715, 0.827080226563316899020244687397, 2.26783321017013087113740577477, 3.56438044831975160604161727351, 5.10041734659472347135761022519, 6.17149663149685483173968852649, 7.10357285017847022325745610366, 7.57787427036713362736842560849, 8.592385072650834456615651409547, 9.047948584938038971375826432464, 10.33012856381968463854717513632

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