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Properties

Label 2-425-85.9-c1-0-19
Degree $2$
Conductor $425$
Sign $0.311 + 0.950i$
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  + (0.213 + 0.213i)2-s + (0.406 − 0.980i)3-s − 1.90i·4-s + (0.295 − 0.122i)6-s + (2.31 − 0.960i)7-s + (0.833 − 0.833i)8-s + (1.32 + 1.32i)9-s + (2.25 − 0.935i)11-s + (−1.87 − 0.775i)12-s − 5.61·13-s + (0.699 + 0.289i)14-s − 3.46·16-s + (−3.06 + 2.76i)17-s + 0.565i·18-s + (5.04 − 5.04i)19-s + ⋯
L(s)  = 1  + (0.150 + 0.150i)2-s + (0.234 − 0.565i)3-s − 0.954i·4-s + (0.120 − 0.0500i)6-s + (0.876 − 0.363i)7-s + (0.294 − 0.294i)8-s + (0.441 + 0.441i)9-s + (0.681 − 0.282i)11-s + (−0.540 − 0.223i)12-s − 1.55·13-s + (0.186 + 0.0774i)14-s − 0.865·16-s + (−0.742 + 0.669i)17-s + 0.133i·18-s + (1.15 − 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41793 - 1.02749i\)
\(L(\frac12)\) \(\approx\) \(1.41793 - 1.02749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.06 - 2.76i)T \)
good2 \( 1 + (-0.213 - 0.213i)T + 2iT^{2} \)
3 \( 1 + (-0.406 + 0.980i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.31 + 0.960i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.25 + 0.935i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.61T + 13T^{2} \)
19 \( 1 + (-5.04 + 5.04i)T - 19iT^{2} \)
23 \( 1 + (0.329 + 0.795i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.43 + 3.46i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.07 + 0.860i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.95 - 4.71i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-4.72 - 11.4i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.85 - 1.85i)T - 43iT^{2} \)
47 \( 1 - 2.30T + 47T^{2} \)
53 \( 1 + (1.96 + 1.96i)T + 53iT^{2} \)
59 \( 1 + (-5.26 - 5.26i)T + 59iT^{2} \)
61 \( 1 + (0.346 + 0.837i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.69iT - 67T^{2} \)
71 \( 1 + (-0.222 - 0.0921i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.96 - 2.47i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (13.5 - 5.62i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-9.82 - 9.82i)T + 83iT^{2} \)
89 \( 1 + 0.395iT - 89T^{2} \)
97 \( 1 + (-8.27 - 3.42i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.05831632188729266773077766933, −10.07030897947089227523392818422, −9.298239141093959752963958556763, −8.048751565929506369381563468743, −7.21670135999308982751842630652, −6.44037685110344236105411594814, −5.03761839333787102997029546796, −4.46612405859092502226924571866, −2.39369280400300019253198296057, −1.19706079215860756906019947736, 2.08250728453056177964178699718, 3.43752466387317227300027843794, 4.41289043220806587840597250690, 5.25614186155763237227050512593, 7.03382363699504615856963758148, 7.58223972305650259130485899008, 8.813270290829548979220802019859, 9.407979868451959299463439137587, 10.42404437470998069232173110464, 11.66652278132403460604044852209

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