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Properties

Label 2-425-17.2-c1-0-17
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: $\chi_{425} (376, \cdot )$
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

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

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