↑

Properties

Label 2-425-17.2-c1-0-13
Degree $2$
Conductor $425$
Sign $-0.209 + 0.977i$
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  + (0.392 + 0.392i)2-s + (−2.31 + 0.958i)3-s − 1.69i·4-s + (−1.28 − 0.532i)6-s + (−1.54 + 3.72i)7-s + (1.45 − 1.45i)8-s + (2.31 − 2.31i)9-s + (−0.388 − 0.160i)11-s + (1.62 + 3.91i)12-s − 4.96i·13-s + (−2.07 + 0.858i)14-s − 2.24·16-s + (0.676 − 4.06i)17-s + 1.81·18-s + (−3.56 − 3.56i)19-s + ⋯
L(s)  = 1  + (0.277 + 0.277i)2-s + (−1.33 + 0.553i)3-s − 0.845i·4-s + (−0.524 − 0.217i)6-s + (−0.583 + 1.40i)7-s + (0.512 − 0.512i)8-s + (0.770 − 0.770i)9-s + (−0.117 − 0.0485i)11-s + (0.467 + 1.12i)12-s − 1.37i·13-s + (−0.553 + 0.229i)14-s − 0.560·16-s + (0.164 − 0.986i)17-s + 0.427·18-s + (−0.817 − 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.209 + 0.977i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.209 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277415 - 0.343230i\)
\(L(\frac12)\) \(\approx\) \(0.277415 - 0.343230i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.676 + 4.06i)T \)
good2 \( 1 + (-0.392 - 0.392i)T + 2iT^{2} \)
3 \( 1 + (2.31 - 0.958i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.54 - 3.72i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.388 + 0.160i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.96iT - 13T^{2} \)
19 \( 1 + (3.56 + 3.56i)T + 19iT^{2} \)
23 \( 1 + (7.44 + 3.08i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.0710 + 0.171i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.15 + 1.30i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-3.52 + 1.46i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.49 - 3.61i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.64 - 3.64i)T - 43iT^{2} \)
47 \( 1 - 6.56iT - 47T^{2} \)
53 \( 1 + (0.616 + 0.616i)T + 53iT^{2} \)
59 \( 1 + (-10.1 + 10.1i)T - 59iT^{2} \)
61 \( 1 + (-1.30 + 3.15i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 7.21T + 67T^{2} \)
71 \( 1 + (6.91 - 2.86i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.37 + 3.31i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (13.0 + 5.38i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-9.66 - 9.66i)T + 83iT^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (3.88 + 9.38i)T + (-68.5 + 68.5i)T^{2} \)
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

−10.91589872585281145395638828853, −10.06936082742225571433202756486, −9.498067057291345847051429706579, −8.198359074314996428352868358481, −6.58946005285369288172120259089, −5.96043833019327044272499155189, −5.35180946589548346677194519154, −4.51601485314832559761020369348, −2.64049591438848798792472686972, −0.29548133722556982024477923190, 1.73663167123277917331861910310, 3.77396365253067567946449859665, 4.36742025342853272684269077944, 5.91068977145071595094012526823, 6.76096620236370706363451016166, 7.44036672998628270670515243398, 8.518675066996283767933019198382, 10.09671394016019484474576609775, 10.65562035571181560415556043395, 11.77129534373019466580256268126

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