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Properties

Label 2-425-85.84-c3-0-48
Degree $2$
Conductor $425$
Sign $-0.216 + 0.976i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 8·3-s + 7·4-s − 8i·6-s − 14·7-s + 15i·8-s + 37·9-s − 20i·11-s − 56·12-s + 58i·13-s − 14i·14-s + 41·16-s + (68 − 17i)17-s + 37i·18-s − 80·19-s + ⋯
L(s)  = 1  + 0.353i·2-s − 1.53·3-s + 0.875·4-s − 0.544i·6-s − 0.755·7-s + 0.662i·8-s + 1.37·9-s − 0.548i·11-s − 1.34·12-s + 1.23i·13-s − 0.267i·14-s + 0.640·16-s + (0.970 − 0.242i)17-s + 0.484i·18-s − 0.965·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.216 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3741054344\)
\(L(\frac12)\) \(\approx\) \(0.3741054344\)
\(L(\frac{5}{2})\) 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 \( 1 + (-68 + 17i)T \)
good2 \( 1 - iT - 8T^{2} \)
3 \( 1 + 8T + 27T^{2} \)
7 \( 1 + 14T + 343T^{2} \)
11 \( 1 + 20iT - 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
19 \( 1 + 80T + 6.85e3T^{2} \)
23 \( 1 + 118T + 1.21e4T^{2} \)
29 \( 1 - 126iT - 2.43e4T^{2} \)
31 \( 1 + 70iT - 2.97e4T^{2} \)
37 \( 1 + 134T + 5.06e4T^{2} \)
41 \( 1 - 100iT - 6.89e4T^{2} \)
43 \( 1 + 272iT - 7.95e4T^{2} \)
47 \( 1 + 464iT - 1.03e5T^{2} \)
53 \( 1 + 642iT - 1.48e5T^{2} \)
59 \( 1 - 180T + 2.05e5T^{2} \)
61 \( 1 + 110iT - 2.26e5T^{2} \)
67 \( 1 + 924iT - 3.00e5T^{2} \)
71 \( 1 + 90iT - 3.57e5T^{2} \)
73 \( 1 + 828T + 3.89e5T^{2} \)
79 \( 1 + 1.33e3iT - 4.93e5T^{2} \)
83 \( 1 + 552iT - 5.71e5T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 - 1.37e3T + 9.12e5T^{2} \)
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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

−10.61309647394231599820996199012, −9.956245242154209799435762412300, −8.593043546861774463613604128801, −7.28381864162805249318513822867, −6.49523364766233869688273100759, −6.01124231391813385895536605858, −5.02444693701406433432415833681, −3.56259258782748810347260999421, −1.83971311097961428864213452523, −0.15722913712257133124419387649, 1.19789397534352385910691848825, 2.80076432898675624744932301579, 4.16510857777031907530480067285, 5.63334233584287910248058534642, 6.12264425848617884042279752055, 7.03725385481933540366396049317, 8.026934644918999633001956108029, 9.836178569640224529499373716024, 10.29426239868705078187495514854, 10.98331427895250149213930783382

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