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Properties

Label 2-425-17.9-c1-0-21
Degree $2$
Conductor $425$
Sign $0.498 + 0.866i$
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  + (1.52 − 1.52i)2-s + (2.56 + 1.06i)3-s − 2.67i·4-s + (5.54 − 2.29i)6-s + (−1.20 − 2.90i)7-s + (−1.03 − 1.03i)8-s + (3.32 + 3.32i)9-s + (−4.70 + 1.94i)11-s + (2.84 − 6.86i)12-s − 1.39i·13-s + (−6.27 − 2.60i)14-s + 2.18·16-s + (1.88 + 3.66i)17-s + 10.1·18-s + (−3.63 + 3.63i)19-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)2-s + (1.48 + 0.613i)3-s − 1.33i·4-s + (2.26 − 0.937i)6-s + (−0.454 − 1.09i)7-s + (−0.366 − 0.366i)8-s + (1.10 + 1.10i)9-s + (−1.41 + 0.587i)11-s + (0.820 − 1.98i)12-s − 0.386i·13-s + (−1.67 − 0.695i)14-s + 0.545·16-s + (0.457 + 0.889i)17-s + 2.39·18-s + (−0.833 + 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.498 + 0.866i$
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: \((2,\ 425,\ (\ :1/2),\ 0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93699 - 1.69935i\)
\(L(\frac12)\) \(\approx\) \(2.93699 - 1.69935i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 3.66i)T \)
good2 \( 1 + (-1.52 + 1.52i)T - 2iT^{2} \)
3 \( 1 + (-2.56 - 1.06i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (1.20 + 2.90i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.70 - 1.94i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.39iT - 13T^{2} \)
19 \( 1 + (3.63 - 3.63i)T - 19iT^{2} \)
23 \( 1 + (5.73 - 2.37i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.65 + 6.41i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.33 - 0.966i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-7.17 - 2.97i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.83 + 9.25i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.08 - 2.08i)T + 43iT^{2} \)
47 \( 1 - 5.08iT - 47T^{2} \)
53 \( 1 + (2.93 - 2.93i)T - 53iT^{2} \)
59 \( 1 + (0.594 + 0.594i)T + 59iT^{2} \)
61 \( 1 + (2.29 + 5.54i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 + (9.66 + 4.00i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.227 + 0.549i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-6.77 + 2.80i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (3.35 - 3.35i)T - 83iT^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + (0.982 - 2.37i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.71142013830741173079027889473, −10.22447530949282378912407207879, −9.781915921105589003208539336218, −8.103070277157143224533277960584, −7.76833672198907050902534227220, −5.93219652493230811516429801131, −4.50714811811844817811463709014, −3.91632389038387380162978453779, −3.02439072199691285889172416343, −2.02030000001154505980109290937, 2.49215976304258972414358080270, 3.15332756918565804071400111970, 4.61324326509953211637556921968, 5.73083647057133596791943733120, 6.63779332091310128108861367185, 7.60701100096535473842115078354, 8.311283018034401596626027805479, 9.045063635833070336612657969881, 10.21547476415089819870861259544, 11.82349585727869429573269250477

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