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Properties

Label 2-425-25.6-c1-0-20
Degree $2$
Conductor $425$
Sign $0.968 + 0.248i$
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  + (−0.118 − 0.363i)2-s + (0.809 − 0.587i)3-s + (1.5 − 1.08i)4-s + (−0.690 + 2.12i)5-s + (−0.309 − 0.224i)6-s + 4.61·7-s + (−1.19 − 0.865i)8-s + (−0.618 + 1.90i)9-s + 0.854·10-s + (1 + 3.07i)11-s + (0.572 − 1.76i)12-s + (−0.572 + 1.76i)13-s + (−0.545 − 1.67i)14-s + (0.690 + 2.12i)15-s + (0.972 − 2.99i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.0834 − 0.256i)2-s + (0.467 − 0.339i)3-s + (0.750 − 0.544i)4-s + (−0.309 + 0.951i)5-s + (−0.126 − 0.0916i)6-s + 1.74·7-s + (−0.421 − 0.305i)8-s + (−0.206 + 0.634i)9-s + 0.270·10-s + (0.301 + 0.927i)11-s + (0.165 − 0.509i)12-s + (−0.158 + 0.489i)13-s + (−0.145 − 0.448i)14-s + (0.178 + 0.549i)15-s + (0.243 − 0.747i)16-s + (−0.196 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91833 - 0.242341i\)
\(L(\frac12)\) \(\approx\) \(1.91833 - 0.242341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.690 - 2.12i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (0.118 + 0.363i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 4.61T + 7T^{2} \)
11 \( 1 + (-1 - 3.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.572 - 1.76i)T + (-10.5 - 7.64i)T^{2} \)
19 \( 1 + (3.92 + 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.690 - 2.12i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.16 + 3.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (8.28 + 6.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.07 + 9.45i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.763 + 2.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.85T + 43T^{2} \)
47 \( 1 + (2.11 - 1.53i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.04 - 2.21i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.59 + 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.54 + 4.75i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-12.4 - 9.06i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.35 - 4.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.78 - 8.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.69 - 1.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.80 + 3.49i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.381 - 1.17i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.59 - 4.78i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.30375080407499869769801319498, −10.53665178133288090665144739302, −9.420353559993133362300863585213, −8.178728959117295928757251029530, −7.42370891199279964747260988454, −6.75913636821039501952811659279, −5.36188699107157834427828212602, −4.20554004691893689590226961450, −2.38564587474365176566553644526, −1.91424830938483051107285895508, 1.53693611815392051877520411391, 3.18369492812314544036078451150, 4.30702943756398238644774227301, 5.39597174991071528412393606003, 6.58859596811431470209221721653, 7.956282765369217086522941160195, 8.414935570239989674862560741040, 8.880432449792546806275317287137, 10.47903319387965773966219176906, 11.34042304215695423963877103995

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