↑

Properties

Label 2-425-85.9-c1-0-10
Degree $2$
Conductor $425$
Sign $-0.961 + 0.274i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (1.82 + 1.82i)2-s + (−1.17 + 2.84i)3-s + 4.67i·4-s + (−7.35 + 3.04i)6-s + (2.60 − 1.07i)7-s + (−4.89 + 4.89i)8-s + (−4.59 − 4.59i)9-s + (0.616 − 0.255i)11-s + (−13.3 − 5.51i)12-s + 4.34·13-s + (6.72 + 2.78i)14-s − 8.52·16-s + (3.16 − 2.64i)17-s − 16.7i·18-s + (−1.88 + 1.88i)19-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)2-s + (−0.680 + 1.64i)3-s + 2.33i·4-s + (−3.00 + 1.24i)6-s + (0.983 − 0.407i)7-s + (−1.72 + 1.72i)8-s + (−1.53 − 1.53i)9-s + (0.185 − 0.0769i)11-s + (−3.84 − 1.59i)12-s + 1.20·13-s + (1.79 + 0.744i)14-s − 2.13·16-s + (0.767 − 0.640i)17-s − 3.95i·18-s + (−0.433 + 0.433i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.961 + 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323266 - 2.30969i\)
\(L(\frac12)\) \(\approx\) \(0.323266 - 2.30969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-3.16 + 2.64i)T \)
good2 \( 1 + (-1.82 - 1.82i)T + 2iT^{2} \)
3 \( 1 + (1.17 - 2.84i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.60 + 1.07i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.616 + 0.255i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
19 \( 1 + (1.88 - 1.88i)T - 19iT^{2} \)
23 \( 1 + (1.15 + 2.78i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.50 + 3.63i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (6.05 + 2.50i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.16 - 2.82i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.38 + 5.76i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (4.37 - 4.37i)T - 43iT^{2} \)
47 \( 1 - 1.08T + 47T^{2} \)
53 \( 1 + (4.94 + 4.94i)T + 53iT^{2} \)
59 \( 1 + (-0.272 - 0.272i)T + 59iT^{2} \)
61 \( 1 + (4.24 + 10.2i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 + (-4.19 - 1.73i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.43 - 3.08i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (6.31 - 2.61i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + 0.844iT - 89T^{2} \)
97 \( 1 + (10.7 + 4.46i)T + (68.5 + 68.5i)T^{2} \)
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

−11.58047894383180660062963629532, −11.03783977790531417571318640469, −9.945132027417120739432166459895, −8.686504843862235578861037657679, −7.908251044828189049134193899391, −6.54799684570377985170759187498, −5.69813043203554161523511204051, −4.96917140173206118381234775860, −4.15787774372559723683058006612, −3.47417759805971988699440606863, 1.31375504933244177682626235594, 1.94532418727007997510307365532, 3.44457355907258011237683173249, 4.94020273403284746472238440708, 5.73676314723347568752392170583, 6.46888401896565895842925478277, 7.76278272562252308578923670082, 8.846619810592603568880675168392, 10.54964931575446964139625733941, 11.12946666729938205749266145929

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