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Properties

Label 2-425-85.9-c1-0-22
Degree $2$
Conductor $425$
Sign $-0.243 + 0.969i$
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  + (0.292 + 0.292i)2-s + (1 − 2.41i)3-s − 1.82i·4-s + (1 − 0.414i)6-s + (1 − 0.414i)7-s + (1.12 − 1.12i)8-s + (−2.70 − 2.70i)9-s + (−1 + 0.414i)11-s + (−4.41 − 1.82i)12-s + 1.41·13-s + (0.414 + 0.171i)14-s − 3·16-s + (3 + 2.82i)17-s − 1.58i·18-s + (−3.41 + 3.41i)19-s + ⋯
L(s)  = 1  + (0.207 + 0.207i)2-s + (0.577 − 1.39i)3-s − 0.914i·4-s + (0.408 − 0.169i)6-s + (0.377 − 0.156i)7-s + (0.396 − 0.396i)8-s + (−0.902 − 0.902i)9-s + (−0.301 + 0.124i)11-s + (−1.27 − 0.527i)12-s + 0.392·13-s + (0.110 + 0.0458i)14-s − 0.750·16-s + (0.727 + 0.685i)17-s − 0.373i·18-s + (−0.783 + 0.783i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.243 + 0.969i$
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: \((2,\ 425,\ (\ :1/2),\ -0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15471 - 1.48017i\)
\(L(\frac12)\) \(\approx\) \(1.15471 - 1.48017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 2.82i)T \)
good2 \( 1 + (-0.292 - 0.292i)T + 2iT^{2} \)
3 \( 1 + (-1 + 2.41i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1 + 0.414i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1 - 0.414i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
19 \( 1 + (3.41 - 3.41i)T - 19iT^{2} \)
23 \( 1 + (-1.58 - 3.82i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.70 + 4.12i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (3 + 1.24i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.46 - 3.53i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.12 + 7.53i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.41 + 3.41i)T - 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-4.24 - 4.24i)T + 59iT^{2} \)
61 \( 1 + (-3.53 - 8.53i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.82iT - 67T^{2} \)
71 \( 1 + (-12.0 - 5i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (4.94 + 2.05i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.82 + 1.58i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.242 + 0.242i)T + 83iT^{2} \)
89 \( 1 - 9.41iT - 89T^{2} \)
97 \( 1 + (5.94 + 2.46i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.88011837613840512510787890070, −10.09277762495400667169469511826, −8.874200394756401818404148840388, −7.975986757986551377851968997761, −7.21833540647213711024846349012, −6.24854746752052352403603705543, −5.44298970401391964287553690734, −3.92187076162873379641938516713, −2.19900918266107781696633965872, −1.20121589693157902927855301237, 2.57639039715506029730153934450, 3.46139040227468367595713524718, 4.45951147905015743018339951142, 5.21450512299763379390149636949, 6.91273253948972901670792519503, 8.134206742415446450192966580123, 8.732314298738925788404848615859, 9.546067370968706705099870425912, 10.69279241486004725532513321552, 11.17834442829990635821476442828

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