Properties

Label 2-650-65.64-c1-0-10
Degree $2$
Conductor $650$
Sign $0.868 - 0.496i$
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  + 2-s + i·3-s + 4-s + i·6-s + 3·7-s + 8-s + 2·9-s + i·12-s + (−3 − 2i)13-s + 3·14-s + 16-s + 3i·17-s + 2·18-s − 6i·19-s + 3i·21-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.13·7-s + 0.353·8-s + 0.666·9-s + 0.288i·12-s + (−0.832 − 0.554i)13-s + 0.801·14-s + 0.250·16-s + 0.727i·17-s + 0.471·18-s − 1.37i·19-s + 0.654i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52292 + 0.669997i\)
\(L(\frac12)\) \(\approx\) \(2.52292 + 0.669997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
13 \( 1 + (3 + 2i)T \)
good3 \( 1 - iT - 3T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−10.77128747249240268522110685343, −9.919582178195877658614606755822, −8.971911149774787757747833746440, −7.77321445513339314779217604491, −7.18900165627792106120454499034, −5.83743036584630238830744322415, −4.87140494897237862716478860139, −4.36530358712088292472820094178, −3.07387344338991951809724368298, −1.65692696845448470095164131922, 1.47710430472257870417924395030, 2.51421172475490529448825642946, 4.16491894389083767687205950364, 4.81601720941620821500807048744, 5.93614338951729146501397390812, 7.00401701590232774144305270559, 7.61891432149947105942559238352, 8.497338597093787120376714571584, 9.792805258678974609615639773783, 10.59358246500303985706051099521

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