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Properties

Label 2-425-85.59-c1-0-18
Degree $2$
Conductor $425$
Sign $0.0525 + 0.998i$
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  + (0.254 + 0.254i)2-s + (0.0501 + 0.0207i)3-s − 1.87i·4-s + (0.00747 + 0.0180i)6-s + (−0.114 − 0.275i)7-s + (0.985 − 0.985i)8-s + (−2.11 − 2.11i)9-s + (−1.05 − 2.55i)11-s + (0.0388 − 0.0937i)12-s + 1.97·13-s + (0.0411 − 0.0994i)14-s − 3.23·16-s + (−3.94 + 1.21i)17-s − 1.07i·18-s + (1.99 − 1.99i)19-s + ⋯
L(s)  = 1  + (0.180 + 0.180i)2-s + (0.0289 + 0.0119i)3-s − 0.935i·4-s + (0.00305 + 0.00736i)6-s + (−0.0432 − 0.104i)7-s + (0.348 − 0.348i)8-s + (−0.706 − 0.706i)9-s + (−0.319 − 0.770i)11-s + (0.0112 − 0.0270i)12-s + 0.549·13-s + (0.0110 − 0.0265i)14-s − 0.809·16-s + (−0.955 + 0.293i)17-s − 0.254i·18-s + (0.457 − 0.457i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0525 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.0525 + 0.998i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.0525 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951611 - 0.902851i\)
\(L(\frac12)\) \(\approx\) \(0.951611 - 0.902851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.94 - 1.21i)T \)
good2 \( 1 + (-0.254 - 0.254i)T + 2iT^{2} \)
3 \( 1 + (-0.0501 - 0.0207i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.114 + 0.275i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.05 + 2.55i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.97T + 13T^{2} \)
19 \( 1 + (-1.99 + 1.99i)T - 19iT^{2} \)
23 \( 1 + (-6.22 + 2.57i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (4.36 + 1.80i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.15 + 2.79i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-8.70 - 3.60i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.87 + 1.19i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.78 - 5.78i)T - 43iT^{2} \)
47 \( 1 - 1.08T + 47T^{2} \)
53 \( 1 + (1.89 + 1.89i)T + 53iT^{2} \)
59 \( 1 + (-6.47 - 6.47i)T + 59iT^{2} \)
61 \( 1 + (-10.3 + 4.28i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 12.5iT - 67T^{2} \)
71 \( 1 + (2.25 - 5.43i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.0829 + 0.200i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.07 - 9.84i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 - 1.55iT - 89T^{2} \)
97 \( 1 + (-3.43 + 8.28i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.27681941565313073444927046644, −10.04994153139072656927129527173, −9.127585259238981255564851199864, −8.404633410738981806390285409512, −6.94904427860757249166300622432, −6.15762282965401309323667382971, −5.35768593474768821728761823213, −4.11487609426100174270958334881, −2.70726007538409121361645952936, −0.800313957666804294838152199317, 2.18986066084643588735149552867, 3.26622094834855341179222846718, 4.51822495789598897643623183251, 5.49765110828925572360711672116, 6.94326864923970680359645136859, 7.72720619347448773566624679384, 8.623575618358397348916524574011, 9.443068617403172621287342632232, 10.85062019082603177599877995361, 11.31830652972592657715458597227

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