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Properties

Label 2-425-17.9-c1-0-20
Degree $2$
Conductor $425$
Sign $-0.725 + 0.687i$
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  + (1.27 − 1.27i)2-s + (−0.635 − 0.263i)3-s − 1.26i·4-s + (−1.14 + 0.475i)6-s + (−1.66 − 4.01i)7-s + (0.943 + 0.943i)8-s + (−1.78 − 1.78i)9-s + (0.0485 − 0.0200i)11-s + (−0.331 + 0.801i)12-s − 3.02i·13-s + (−7.24 − 3.00i)14-s + 4.93·16-s + (−3.12 + 2.69i)17-s − 4.56·18-s + (5.52 − 5.52i)19-s + ⋯
L(s)  = 1  + (0.902 − 0.902i)2-s + (−0.366 − 0.151i)3-s − 0.630i·4-s + (−0.468 + 0.194i)6-s + (−0.628 − 1.51i)7-s + (0.333 + 0.333i)8-s + (−0.595 − 0.595i)9-s + (0.0146 − 0.00605i)11-s + (−0.0958 + 0.231i)12-s − 0.839i·13-s + (−1.93 − 0.801i)14-s + 1.23·16-s + (−0.757 + 0.653i)17-s − 1.07·18-s + (1.26 − 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.725 + 0.687i$
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: \((2,\ 425,\ (\ :1/2),\ -0.725 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625553 - 1.57003i\)
\(L(\frac12)\) \(\approx\) \(0.625553 - 1.57003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 - 2.69i)T \)
good2 \( 1 + (-1.27 + 1.27i)T - 2iT^{2} \)
3 \( 1 + (0.635 + 0.263i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.66 + 4.01i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.0485 + 0.0200i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
19 \( 1 + (-5.52 + 5.52i)T - 19iT^{2} \)
23 \( 1 + (0.962 - 0.398i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.161 - 0.388i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.27 + 0.529i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.311 + 0.128i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.52 - 6.09i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-7.06 - 7.06i)T + 43iT^{2} \)
47 \( 1 - 6.13iT - 47T^{2} \)
53 \( 1 + (-8.52 + 8.52i)T - 53iT^{2} \)
59 \( 1 + (-3.60 - 3.60i)T + 59iT^{2} \)
61 \( 1 + (2.28 + 5.51i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 0.916T + 67T^{2} \)
71 \( 1 + (-3.86 - 1.59i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.06 - 4.98i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.22 + 3.82i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.61 + 4.61i)T - 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + (-7.35 + 17.7i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.07164096941629662270197244727, −10.33561242449973603397961747309, −9.326872860735001315816675490610, −7.919192003755273912518959671097, −6.96306652704203825673310964235, −5.89330111559224894200072698614, −4.72411616136481710340240999589, −3.70200771270816436363672086225, −2.87862042362169399504507094596, −0.876332064651138356347629992272, 2.39962299581080215231330983059, 3.87024438146246072728442277421, 5.22185621435130803290481599773, 5.64950993538585464937973082184, 6.51473445649481603810691879834, 7.55758929116301521597000052812, 8.757930590463991704572077795810, 9.578139532510977022926736678393, 10.70926544706722907505750388630, 11.94995712253692877330333377035

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