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Properties

Label 2-425-85.19-c1-0-13
Degree $2$
Conductor $425$
Sign $0.956 + 0.290i$
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 + (0.263 + 0.635i)3-s − 1.26i·4-s + (−1.14 − 0.475i)6-s + (−4.01 − 1.66i)7-s + (−0.943 − 0.943i)8-s + (1.78 − 1.78i)9-s + (0.0485 + 0.0200i)11-s + (0.801 − 0.331i)12-s + 3.02·13-s + (7.24 − 3.00i)14-s + 4.93·16-s + (2.69 − 3.12i)17-s + 4.56i·18-s + (−5.52 − 5.52i)19-s + ⋯
L(s)  = 1  + (−0.902 + 0.902i)2-s + (0.151 + 0.366i)3-s − 0.630i·4-s + (−0.468 − 0.194i)6-s + (−1.51 − 0.628i)7-s + (−0.333 − 0.333i)8-s + (0.595 − 0.595i)9-s + (0.0146 + 0.00605i)11-s + (0.231 − 0.0958i)12-s + 0.839·13-s + (1.93 − 0.801i)14-s + 1.23·16-s + (0.653 − 0.757i)17-s + 1.07i·18-s + (−1.26 − 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596926 - 0.0885878i\)
\(L(\frac12)\) \(\approx\) \(0.596926 - 0.0885878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.69 + 3.12i)T \)
good2 \( 1 + (1.27 - 1.27i)T - 2iT^{2} \)
3 \( 1 + (-0.263 - 0.635i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (4.01 + 1.66i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.0485 - 0.0200i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
19 \( 1 + (5.52 + 5.52i)T + 19iT^{2} \)
23 \( 1 + (0.398 - 0.962i)T + (-16.2 - 16.2i)T^{2} \)
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 \( 1 + (0.128 + 0.311i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.52 + 6.09i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (7.06 + 7.06i)T + 43iT^{2} \)
47 \( 1 - 6.13T + 47T^{2} \)
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 \( 1 + 0.916iT - 67T^{2} \)
71 \( 1 + (-3.86 + 1.59i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.98 - 2.06i)T + (51.6 - 51.6i)T^{2} \)
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 \( 1 + (17.7 - 7.35i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.63297291184443698052383837424, −9.963005795898547255144617377009, −9.202538369336335291570939496818, −8.616663812990017442729647378653, −7.12907981808020948862549602673, −6.86602220990260046734125012387, −5.83718046704469167593816775659, −4.07726185987732576156311725866, −3.20267952985319882545919357029, −0.53137029217176557584753675560, 1.51371319316106998721950449636, 2.70776948718538210579684219819, 3.86477428242856416487077516031, 5.79806698355748555518663736784, 6.47154958016097484683235256726, 7.955207445054199430528730457559, 8.615807102365321730023674225986, 9.621454984330430300349507077511, 10.24122612673338283164624946688, 10.92513427175347965930638657007

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