Properties

Label 2-425-425.16-c1-0-9
Degree $2$
Conductor $425$
Sign $0.285 - 0.958i$
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  + (−0.484 − 0.352i)2-s + (−0.406 + 0.131i)3-s + (−0.506 − 1.56i)4-s + (1.58 + 1.58i)5-s + (0.243 + 0.0790i)6-s + 1.28i·7-s + (−0.674 + 2.07i)8-s + (−2.27 + 1.65i)9-s + (−0.210 − 1.32i)10-s + (−1.22 + 1.69i)11-s + (0.411 + 0.566i)12-s + (−1.13 + 0.828i)13-s + (0.452 − 0.622i)14-s + (−0.851 − 0.432i)15-s + (−1.59 + 1.15i)16-s + (3.68 − 1.84i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.249i)2-s + (−0.234 + 0.0761i)3-s + (−0.253 − 0.780i)4-s + (0.707 + 0.706i)5-s + (0.0993 + 0.0322i)6-s + 0.485i·7-s + (−0.238 + 0.733i)8-s + (−0.759 + 0.552i)9-s + (−0.0665 − 0.418i)10-s + (−0.370 + 0.509i)11-s + (0.118 + 0.163i)12-s + (−0.316 + 0.229i)13-s + (0.120 − 0.166i)14-s + (−0.219 − 0.111i)15-s + (−0.399 + 0.289i)16-s + (0.894 − 0.447i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.285 - 0.958i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.285 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657911 + 0.490699i\)
\(L(\frac12)\) \(\approx\) \(0.657911 + 0.490699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 1.58i)T \)
17 \( 1 + (-3.68 + 1.84i)T \)
good2 \( 1 + (0.484 + 0.352i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.406 - 0.131i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 - 1.28iT - 7T^{2} \)
11 \( 1 + (1.22 - 1.69i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.13 - 0.828i)T + (4.01 - 12.3i)T^{2} \)
19 \( 1 + (1.63 - 5.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.00 + 1.38i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (6.37 - 2.07i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.60 - 1.82i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.76 - 7.93i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.01 - 4.14i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 + (1.98 + 6.10i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.388 + 1.19i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.686 + 0.499i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.13 - 1.56i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.69 + 5.21i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (7.27 - 2.36i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.87 + 6.70i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.42 - 1.76i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.29 + 3.99i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.66 + 4.84i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.68 + 2.17i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.22492249130152909103908985663, −10.17407467050154437199241637444, −9.949684425700393707297537799537, −8.824776284217331074340219452691, −7.80659531970731553432640359587, −6.43764094338475418664020880327, −5.63530632074445201999257585851, −4.89795283267512714157630033125, −2.90147093532763534766763895734, −1.83060644921490701730921606776, 0.59885108998863385436857981571, 2.78142351501833269884828089337, 4.06536697232659603135292673307, 5.34896727170138240634441523379, 6.25563658218719158726981849592, 7.43748000349730956458894143694, 8.322217281952073923303575814558, 9.103145517911536142343091161727, 9.836627277903436879328007880185, 11.01987926301837998971458872215

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