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Properties

Label 2-425-85.19-c1-0-19
Degree $2$
Conductor $425$
Sign $-0.103 + 0.994i$
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  + (−1.01 + 1.01i)2-s + (0.0420 + 0.101i)3-s − 0.0689i·4-s + (−0.146 − 0.0604i)6-s + (−0.642 − 0.265i)7-s + (−1.96 − 1.96i)8-s + (2.11 − 2.11i)9-s + (−4.48 − 1.85i)11-s + (0.00700 − 0.00290i)12-s − 5.63·13-s + (0.923 − 0.382i)14-s + 4.13·16-s + (−3.78 + 1.63i)17-s + 4.29i·18-s + (1.64 + 1.64i)19-s + ⋯
L(s)  = 1  + (−0.719 + 0.719i)2-s + (0.0242 + 0.0586i)3-s − 0.0344i·4-s + (−0.0596 − 0.0246i)6-s + (−0.242 − 0.100i)7-s + (−0.694 − 0.694i)8-s + (0.704 − 0.704i)9-s + (−1.35 − 0.559i)11-s + (0.00202 − 0.000837i)12-s − 1.56·13-s + (0.246 − 0.102i)14-s + 1.03·16-s + (−0.918 + 0.395i)17-s + 1.01i·18-s + (0.376 + 0.376i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145314 - 0.161182i\)
\(L(\frac12)\) \(\approx\) \(0.145314 - 0.161182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.78 - 1.63i)T \)
good2 \( 1 + (1.01 - 1.01i)T - 2iT^{2} \)
3 \( 1 + (-0.0420 - 0.101i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.642 + 0.265i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (4.48 + 1.85i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 5.63T + 13T^{2} \)
19 \( 1 + (-1.64 - 1.64i)T + 19iT^{2} \)
23 \( 1 + (-1.77 + 4.28i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (2.48 + 6.01i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (6.12 - 2.53i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.0453 - 0.109i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.412 - 0.996i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.453 - 0.453i)T + 43iT^{2} \)
47 \( 1 - 4.93T + 47T^{2} \)
53 \( 1 + (-8.47 + 8.47i)T - 53iT^{2} \)
59 \( 1 + (7.01 - 7.01i)T - 59iT^{2} \)
61 \( 1 + (-0.613 + 1.48i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 2.99iT - 67T^{2} \)
71 \( 1 + (4.33 - 1.79i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.08 - 2.10i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-13.7 - 5.68i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (3.56 - 3.56i)T - 83iT^{2} \)
89 \( 1 - 2.35iT - 89T^{2} \)
97 \( 1 + (-2.49 + 1.03i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.59410296600920471031617637779, −9.852639649515942507323148999717, −9.072883194925716245165787379412, −8.083179388555371383416762884569, −7.30755407435644973086869627819, −6.56057269790998196560042969932, −5.34401395042997332620078320755, −3.97894715519881704230802107781, −2.64385902654654608276793467138, −0.16107929684353202107636653578, 1.94424620735748239727974549900, 2.80595613289815799064852280165, 4.77042545249802083100582310376, 5.44849917143860735550883245563, 7.18149148325355457094114773984, 7.67772382068474331626489333606, 9.096521629818952873533173907371, 9.667318133769218857590869890024, 10.51097384823496160511461594542, 11.08202041782950347579276828102

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