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Properties

Label 2-425-17.9-c1-0-13
Degree $2$
Conductor $425$
Sign $0.651 - 0.758i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.292 − 0.292i)2-s + (2.41 + i)3-s + 1.82i·4-s + (1 − 0.414i)6-s + (−0.414 − i)7-s + (1.12 + 1.12i)8-s + (2.70 + 2.70i)9-s + (−1 + 0.414i)11-s + (−1.82 + 4.41i)12-s + 1.41i·13-s + (−0.414 − 0.171i)14-s − 3·16-s + (2.82 − 3i)17-s + 1.58·18-s + (3.41 − 3.41i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.207i)2-s + (1.39 + 0.577i)3-s + 0.914i·4-s + (0.408 − 0.169i)6-s + (−0.156 − 0.377i)7-s + (0.396 + 0.396i)8-s + (0.902 + 0.902i)9-s + (−0.301 + 0.124i)11-s + (−0.527 + 1.27i)12-s + 0.392i·13-s + (−0.110 − 0.0458i)14-s − 0.750·16-s + (0.685 − 0.727i)17-s + 0.373·18-s + (0.783 − 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10919 + 0.969052i\)
\(L(\frac12)\) \(\approx\) \(2.10919 + 0.969052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 + 3i)T \)
good2 \( 1 + (-0.292 + 0.292i)T - 2iT^{2} \)
3 \( 1 + (-2.41 - i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.414 + i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1 - 0.414i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (-3.41 + 3.41i)T - 19iT^{2} \)
23 \( 1 + (3.82 - 1.58i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.70 - 4.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (3 + 1.24i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-3.53 - 1.46i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.12 + 7.53i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.41 - 3.41i)T + 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (4.24 + 4.24i)T + 59iT^{2} \)
61 \( 1 + (-3.53 - 8.53i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + (-12.0 - 5i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.05 + 4.94i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.82 - 1.58i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.242 + 0.242i)T - 83iT^{2} \)
89 \( 1 + 9.41iT - 89T^{2} \)
97 \( 1 + (2.46 - 5.94i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.37303913887946705054655621260, −10.20173142887619870191747990436, −9.371444665787621601162911346146, −8.654500064107235250136173197101, −7.68728759420289416510957159272, −7.11013807733971123937598147497, −5.17482951676055346458983881819, −4.01106211499401092237501835915, −3.30907651229593327510392918745, −2.30711284092580267515076825826, 1.48978440265783511149900612478, 2.71986756619836849882898427047, 3.95365982836172362509516830293, 5.49026822161374889980868311400, 6.28702476867377199079320141231, 7.59555261752939023237390525775, 8.130638148787462970269261014287, 9.280633543800656593680385182499, 9.900719555642209806992789827211, 10.89375273319578653856606563348

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