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Properties

Label 2-650-65.9-c1-0-7
Degree $2$
Conductor $650$
Sign $0.888 - 0.458i$
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 + (1.73 + i)3-s + (0.499 + 0.866i)4-s + (−0.999 − 1.73i)6-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s + 1.99i·12-s + (2.59 + 2.5i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (5.19 − 3i)17-s − 0.999i·18-s + (2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.999 + 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.408 − 0.707i)6-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s + 0.577i·12-s + (0.720 + 0.693i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (1.26 − 0.727i)17-s − 0.235i·18-s + (0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.888 - 0.458i$
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: \((2,\ 650,\ (\ :1/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58737 + 0.385489i\)
\(L(\frac12)\) \(\approx\) \(1.58737 + 0.385489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 2.5i)T \)
good3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.28220314519950221602158461473, −9.680558271270315461127628451295, −9.104005917387731026724803026216, −7.990361585480651687916223845462, −7.64168131335912765919190512963, −6.25656287766773930977753511707, −4.84032604368685439769706597810, −3.75607713291279916733725790841, −2.89106132255136290956354774403, −1.52670808408396028955458847836, 1.13560295675660000694273358100, 2.53851024791436514512038992309, 3.52200574949195190980286740944, 5.30589927153338176698361575158, 6.04940085515767421571001140015, 7.42950705454366284363466247234, 7.87016404945340496765988811010, 8.630052536884357810449586131923, 9.258067027856122314113928544519, 10.47671968980934390338728128105

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