↑

Properties

Label 2-425-17.10-c2-0-51
Degree $2$
Conductor $425$
Sign $-0.857 - 0.513i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 3.34i)2-s + (2.91 + 1.95i)3-s + (−6.43 − 6.43i)4-s + (10.5 − 7.05i)6-s + (−10.4 − 2.08i)7-s + (−17.0 + 7.05i)8-s + (1.27 + 3.07i)9-s + (−9.52 − 14.2i)11-s + (−6.23 − 31.3i)12-s + (−7.87 + 7.87i)13-s + (−21.4 + 32.0i)14-s + 30.3i·16-s + (16.2 + 5.01i)17-s + 12.0·18-s + (7.56 − 18.2i)19-s + ⋯
L(s)  = 1  + (0.692 − 1.67i)2-s + (0.973 + 0.650i)3-s + (−1.60 − 1.60i)4-s + (1.76 − 1.17i)6-s + (−1.49 − 0.297i)7-s + (−2.13 + 0.882i)8-s + (0.141 + 0.341i)9-s + (−0.865 − 1.29i)11-s + (−0.519 − 2.61i)12-s + (−0.605 + 0.605i)13-s + (−1.53 + 2.29i)14-s + 1.89i·16-s + (0.955 + 0.294i)17-s + 0.668·18-s + (0.397 − 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.857 - 0.513i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ -0.857 - 0.513i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.451351 + 1.63188i\)
\(L(\frac12)\) \(\approx\) \(0.451351 + 1.63188i\)
\(L(2)\) 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 \( 1 + (-16.2 - 5.01i)T \)
good2 \( 1 + (-1.38 + 3.34i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (-2.91 - 1.95i)T + (3.44 + 8.31i)T^{2} \)
7 \( 1 + (10.4 + 2.08i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (9.52 + 14.2i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (7.87 - 7.87i)T - 169iT^{2} \)
19 \( 1 + (-7.56 + 18.2i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (2.60 - 1.74i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (4.61 + 23.2i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (4.79 - 7.17i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (-43.3 - 28.9i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (2.48 + 0.493i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (10.9 + 26.4i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (14.1 - 14.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (8.84 - 21.3i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (0.450 - 0.186i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-20.2 + 101. i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 + 70.2iT - 4.48e3T^{2} \)
71 \( 1 + (-44.3 - 29.6i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (-122. + 24.3i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (-23.3 - 34.9i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (75.2 + 31.1i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-1.81 - 1.81i)T + 7.92e3iT^{2} \)
97 \( 1 + (-32.1 - 161. i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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

−10.39208997506508646441078680075, −9.637713563717135480133945945951, −9.300532591755168112985190404682, −8.047194482057289137341095816357, −6.33240741598495091158287465713, −5.12799238313437554399563903307, −3.88076059508573117448986020661, −3.20859869850230177905767912490, −2.58860547765481362692374270194, −0.48608343152157020291635242650, 2.63608200481413072659094695669, 3.59709328122856910740272193578, 5.08399213318006140759142395775, 5.91531136216987326036439422635, 7.09142859165141786442252976851, 7.52804572733237602849839802547, 8.267961652819745998652652536469, 9.426432278582570276339844893627, 10.06400677888133970611016175427, 12.29345323652970186533683097234

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