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Properties

Label 2-650-65.64-c1-0-13
Degree $2$
Conductor $650$
Sign $0.517 + 0.855i$
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 + 2.88i·3-s + 4-s − 2.88i·6-s − 1.88·7-s − 8-s − 5.30·9-s − 6.18i·11-s + 2.88i·12-s + (−0.287 − 3.59i)13-s + 1.88·14-s + 16-s − 3i·17-s + 5.30·18-s − 4.88i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.66i·3-s + 0.5·4-s − 1.17i·6-s − 0.711·7-s − 0.353·8-s − 1.76·9-s − 1.86i·11-s + 0.831i·12-s + (−0.0798 − 0.996i)13-s + 0.502·14-s + 0.250·16-s − 0.727i·17-s + 1.25·18-s − 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.517 + 0.855i$
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: \((2,\ 650,\ (\ :1/2),\ 0.517 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416433 - 0.234905i\)
\(L(\frac12)\) \(\approx\) \(0.416433 - 0.234905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.287 + 3.59i)T \)
good3 \( 1 - 2.88iT - 3T^{2} \)
7 \( 1 + 1.88T + 7T^{2} \)
11 \( 1 + 6.18iT - 11T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 4.88iT - 19T^{2} \)
23 \( 1 - 0.575iT - 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 - 7.30iT - 31T^{2} \)
37 \( 1 + 9.18T + 37T^{2} \)
41 \( 1 - 5.45iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2.45T + 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 + 8.83iT - 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 - 3.57T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 0.188T + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−10.30593573271239788476789546679, −9.572721207825303424023736093307, −8.879369162841929075395870714644, −8.235770529807241089064460726880, −6.83764047620476510870122705456, −5.72623990610386072389577898691, −4.99531556037090644981296759119, −3.39223453000705999914004291554, −3.08032689307719554541300912809, −0.31263851689319016462841469547, 1.63365515882574958306698841059, 2.26110084197248033358103984985, 3.97203127952143496534761714050, 5.72264007083583105772969278013, 6.62061578213497195449741880913, 7.21699954051959938848111538106, 7.81946245098516292869443958876, 8.907989244956390301468751220473, 9.708402626508777554600127549833, 10.58982464450652573968852044936

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