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Properties

Label 2-425-17.16-c1-0-25
Degree $2$
Conductor $425$
Sign $-0.765 + 0.643i$
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  + 1.48·2-s − 3.29i·3-s + 0.193·4-s − 4.87i·6-s − 2.22i·7-s − 2.67·8-s − 7.83·9-s + 2.86i·11-s − 0.638i·12-s + 2.28·13-s − 3.29i·14-s − 4.35·16-s + (3.15 − 2.65i)17-s − 11.5·18-s + 5.76·19-s + ⋯
L(s)  = 1  + 1.04·2-s − 1.90i·3-s + 0.0969·4-s − 1.99i·6-s − 0.839i·7-s − 0.945·8-s − 2.61·9-s + 0.862i·11-s − 0.184i·12-s + 0.634·13-s − 0.879i·14-s − 1.08·16-s + (0.765 − 0.643i)17-s − 2.73·18-s + 1.32·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639088 - 1.75366i\)
\(L(\frac12)\) \(\approx\) \(0.639088 - 1.75366i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.15 + 2.65i)T \)
good2 \( 1 - 1.48T + 2T^{2} \)
3 \( 1 + 3.29iT - 3T^{2} \)
7 \( 1 + 2.22iT - 7T^{2} \)
11 \( 1 - 2.86iT - 11T^{2} \)
13 \( 1 - 2.28T + 13T^{2} \)
19 \( 1 - 5.76T + 19T^{2} \)
23 \( 1 + 1.58iT - 23T^{2} \)
29 \( 1 + 9.23iT - 29T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 + 0.514iT - 37T^{2} \)
41 \( 1 + 7.09iT - 41T^{2} \)
43 \( 1 + 7.89T + 43T^{2} \)
47 \( 1 - 3.03T + 47T^{2} \)
53 \( 1 - 5.73T + 53T^{2} \)
59 \( 1 + 7.50T + 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 - 7.35T + 67T^{2} \)
71 \( 1 - 8.80iT - 71T^{2} \)
73 \( 1 - 2.65iT - 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 - 2.15T + 89T^{2} \)
97 \( 1 + 8.72iT - 97T^{2} \)
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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

−11.47609297996199021195190339686, −9.963663646947963552144440333679, −8.744962564256339881819426112396, −7.64203025291849273891202041040, −7.06994747418436006377180009572, −6.08538601996168966526959932757, −5.20037599032290634139940591865, −3.75554655579716091157645102437, −2.52046630249875102085205311784, −0.912105174608685865190191561326, 3.23209841217578580637727543108, 3.46657527955908020102966059679, 4.81725843405412052491277975555, 5.47778936992801384962393316123, 6.10133655695673974867401004248, 8.284204554202767961512558465371, 9.001998240742008648035340390595, 9.672668978451985835844495248360, 10.73523495519081327458972201026, 11.52284607153626153384823842564

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