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Properties

Label 2-425-17.10-c2-0-34
Degree $2$
Conductor $425$
Sign $0.939 + 0.343i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.885 − 2.13i)2-s + (4.16 + 2.78i)3-s + (−0.954 − 0.954i)4-s + (9.62 − 6.43i)6-s + (−0.970 − 0.193i)7-s + (5.66 − 2.34i)8-s + (6.14 + 14.8i)9-s + (0.201 + 0.301i)11-s + (−1.31 − 6.62i)12-s + (4.39 − 4.39i)13-s + (−1.27 + 1.90i)14-s − 19.5i·16-s + (16.8 + 1.86i)17-s + 37.1·18-s + (−9.20 + 22.2i)19-s + ⋯
L(s)  = 1  + (0.442 − 1.06i)2-s + (1.38 + 0.926i)3-s + (−0.238 − 0.238i)4-s + (1.60 − 1.07i)6-s + (−0.138 − 0.0275i)7-s + (0.707 − 0.293i)8-s + (0.682 + 1.64i)9-s + (0.0183 + 0.0274i)11-s + (−0.109 − 0.552i)12-s + (0.338 − 0.338i)13-s + (−0.0908 + 0.135i)14-s − 1.22i·16-s + (0.993 + 0.109i)17-s + 2.06·18-s + (−0.484 + 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ 0.939 + 0.343i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.68298 - 0.652030i\)
\(L(\frac12)\) \(\approx\) \(3.68298 - 0.652030i\)
\(L(2)\) 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 \( 1 + (-16.8 - 1.86i)T \)
good2 \( 1 + (-0.885 + 2.13i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (-4.16 - 2.78i)T + (3.44 + 8.31i)T^{2} \)
7 \( 1 + (0.970 + 0.193i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (-0.201 - 0.301i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (-4.39 + 4.39i)T - 169iT^{2} \)
19 \( 1 + (9.20 - 22.2i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (21.5 - 14.4i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (8.40 + 42.2i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (-9.36 + 14.0i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (17.9 + 12.0i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (-67.7 - 13.4i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (-2.12 - 5.12i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (60.9 - 60.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (6.20 - 14.9i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-60.5 + 25.0i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-2.42 + 12.1i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 + 39.4iT - 4.48e3T^{2} \)
71 \( 1 + (65.4 + 43.7i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (-0.347 + 0.0692i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (44.3 + 66.3i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (79.3 + 32.8i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (120. + 120. i)T + 7.92e3iT^{2} \)
97 \( 1 + (-17.2 - 86.7i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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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

−10.78236536679074358936439184410, −9.917314032791620924465613549435, −9.592877222163549091828176781045, −8.138342253442482129895682265769, −7.74412824428667230903037251276, −5.89574859122293611795483882588, −4.39070777189745474284425136385, −3.72863098322891005237160187657, −2.92050293912616876957534089555, −1.77340738246629389283214507139, 1.52163738431668723502578028929, 2.81123460814806067028330675801, 4.11345920029663619302507045563, 5.48788938409516591742157231830, 6.71594645967084014960504690360, 7.10966002992510909014322489096, 8.183107549120362251254704761196, 8.673207070424072844506356036370, 9.830079028949241285148977462043, 11.04620702736883346248862795579

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