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Properties

Label 2-425-25.21-c1-0-26
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: $\chi_{425} (171, \cdot )$
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

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

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