↑

Properties

Label 2-650-65.64-c1-0-15
Degree $2$
Conductor $650$
Sign $-0.679 + 0.733i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.10i·3-s + 4-s + 3.10i·6-s + 4.10·7-s − 8-s − 6.64·9-s − 1.53i·11-s − 3.10i·12-s + (3.37 − 1.26i)13-s − 4.10·14-s + 16-s − 3i·17-s + 6.64·18-s + 1.10i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.79i·3-s + 0.5·4-s + 1.26i·6-s + 1.55·7-s − 0.353·8-s − 2.21·9-s − 0.463i·11-s − 0.896i·12-s + (0.935 − 0.352i)13-s − 1.09·14-s + 0.250·16-s − 0.727i·17-s + 1.56·18-s + 0.253i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.679 + 0.733i$
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: \((2,\ 650,\ (\ :1/2),\ -0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486034 - 1.11310i\)
\(L(\frac12)\) \(\approx\) \(0.486034 - 1.11310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.37 + 1.26i)T \)
good3 \( 1 + 3.10iT - 3T^{2} \)
7 \( 1 - 4.10T + 7T^{2} \)
11 \( 1 + 1.53iT - 11T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 1.10iT - 19T^{2} \)
23 \( 1 + 6.74iT - 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 - 8.64iT - 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 0.433iT - 53T^{2} \)
59 \( 1 - 13.7iT - 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 3.74T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 - 4.46T + 83T^{2} \)
89 \( 1 - 1.85iT - 89T^{2} \)
97 \( 1 + 3.78T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.55548412064494475507797870333, −8.748430372543159106688707892872, −8.439872953591965742027394247661, −7.73166099885854251656047432197, −6.87533945692530101753134202991, −6.07227418559170845867135145854, −4.96621917570163232772085458609, −2.97474790498175009554043921491, −1.77859796780718848558916974733, −0.903845166584992108886098823729, 1.77468728970257667617167050330, 3.43123729708044969637221129068, 4.42827005798387104995198927036, 5.18725115539176659199629696688, 6.26591697288867019438187009254, 7.85017287609683636269354406212, 8.404914945757337497951234867368, 9.281477166382507000611069562766, 9.934152169832102945991429135548, 10.89161127905485905050468508667

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