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Properties

Label 2-425-17.8-c1-0-18
Degree $2$
Conductor $425$
Sign $0.202 + 0.979i$
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 + (0.703 − 1.69i)3-s + 0.318i·4-s + (−0.912 − 2.20i)6-s + (3.23 − 1.34i)7-s + (2.12 + 2.12i)8-s + (−0.267 − 0.267i)9-s + (−1.46 − 3.53i)11-s + (0.540 + 0.223i)12-s + 3.99i·13-s + (1.74 − 4.20i)14-s + 3.26·16-s + (−3.26 − 2.51i)17-s − 0.490·18-s + (−3.98 + 3.98i)19-s + ⋯
L(s)  = 1  + (0.648 − 0.648i)2-s + (0.406 − 0.980i)3-s + 0.159i·4-s + (−0.372 − 0.899i)6-s + (1.22 − 0.507i)7-s + (0.751 + 0.751i)8-s + (−0.0890 − 0.0890i)9-s + (−0.441 − 1.06i)11-s + (0.155 + 0.0645i)12-s + 1.10i·13-s + (0.465 − 1.12i)14-s + 0.815·16-s + (−0.791 − 0.611i)17-s − 0.115·18-s + (−0.914 + 0.914i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.202 + 0.979i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87103 - 1.52381i\)
\(L(\frac12)\) \(\approx\) \(1.87103 - 1.52381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.26 + 2.51i)T \)
good2 \( 1 + (-0.917 + 0.917i)T - 2iT^{2} \)
3 \( 1 + (-0.703 + 1.69i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-3.23 + 1.34i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.46 + 3.53i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.99iT - 13T^{2} \)
19 \( 1 + (3.98 - 3.98i)T - 19iT^{2} \)
23 \( 1 + (0.960 + 2.31i)T + (-16.2 + 16.2i)T^{2} \)
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 \( 1 + (0.738 - 1.78i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.730 + 0.302i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.704 + 0.704i)T + 43iT^{2} \)
47 \( 1 - 9.47iT - 47T^{2} \)
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 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + (2.09 - 5.05i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-6.06 - 2.51i)T + (51.6 + 51.6i)T^{2} \)
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 \( 1 + (-13.1 - 5.45i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.20539135737785323993741333614, −10.57380604494175695455133070661, −8.879235759176682031138395967763, −8.034153224600675844499443758531, −7.53829274740722314092032068606, −6.30160582940336657979660759853, −4.81164477107915965901871600435, −4.02171896808133349852694971174, −2.50802397294472696149113482115, −1.62695296789986990450615901376, 2.03267043519620740400746993749, 3.79933671210113684534049423942, 4.83007943623984144753203586131, 5.21236103081912370802759496890, 6.56935850481242385844671429491, 7.66529363056782342429832217190, 8.615633978332362844022386791516, 9.608209359799181405577387368986, 10.49473950122938178809872613052, 11.07827578580907123619555283819

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