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Properties

Label 2-425-85.49-c1-0-10
Degree $2$
Conductor $425$
Sign $0.966 - 0.256i$
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  + (0.917 − 0.917i)2-s + (−1.69 + 0.703i)3-s + 0.318i·4-s + (−0.912 + 2.20i)6-s + (1.34 − 3.23i)7-s + (2.12 + 2.12i)8-s + (0.267 − 0.267i)9-s + (−1.46 + 3.53i)11-s + (−0.223 − 0.540i)12-s + 3.99·13-s + (−1.74 − 4.20i)14-s + 3.26·16-s + (2.51 + 3.26i)17-s − 0.490i·18-s + (3.98 + 3.98i)19-s + ⋯
L(s)  = 1  + (0.648 − 0.648i)2-s + (−0.980 + 0.406i)3-s + 0.159i·4-s + (−0.372 + 0.899i)6-s + (0.507 − 1.22i)7-s + (0.751 + 0.751i)8-s + (0.0890 − 0.0890i)9-s + (−0.441 + 1.06i)11-s + (−0.0645 − 0.155i)12-s + 1.10·13-s + (−0.465 − 1.12i)14-s + 0.815·16-s + (0.611 + 0.791i)17-s − 0.115i·18-s + (0.914 + 0.914i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51849 + 0.198371i\)
\(L(\frac12)\) \(\approx\) \(1.51849 + 0.198371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 - 3.26i)T \)
good2 \( 1 + (-0.917 + 0.917i)T - 2iT^{2} \)
3 \( 1 + (1.69 - 0.703i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.34 + 3.23i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.46 - 3.53i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.99T + 13T^{2} \)
19 \( 1 + (-3.98 - 3.98i)T + 19iT^{2} \)
23 \( 1 + (2.31 + 0.960i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-5.14 + 2.12i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.843 - 2.03i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.78 - 0.738i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.730 - 0.302i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.704 + 0.704i)T + 43iT^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + (-3.87 + 3.87i)T - 53iT^{2} \)
59 \( 1 + (2.12 - 2.12i)T - 59iT^{2} \)
61 \( 1 + (12.9 + 5.37i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 15.9iT - 67T^{2} \)
71 \( 1 + (2.09 + 5.05i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.51 - 6.06i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.54 + 10.9i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.8 - 10.8i)T - 83iT^{2} \)
89 \( 1 - 4.92iT - 89T^{2} \)
97 \( 1 + (5.45 + 13.1i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.24981115012732815385785236781, −10.47951175107490237945124433878, −10.09405987479042656674953283167, −8.212896208170197532738940381994, −7.62817438683023094113087115598, −6.25316027219527117612485628550, −5.08480855459943121032757760288, −4.37161194771167583268174266613, −3.43646001515816764822894419101, −1.57497881900485251946114063693, 1.06799412543329883342493773436, 3.11346120166255624884036300835, 4.86584022223957655553647999204, 5.64157435812079741264051642404, 6.01669727344474931532233097160, 7.06734631593927198039367865954, 8.258601369760602607877352529715, 9.210970143951580637856538420917, 10.51011601853258556472708642573, 11.44108254750162680389589301203

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