Properties

Label 2-425-5.4-c3-0-44
Degree $2$
Conductor $425$
Sign $0.894 + 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.50i·2-s − 5.08i·3-s − 12.2·4-s + 22.9·6-s − 0.616i·7-s − 19.3i·8-s + 1.09·9-s − 8.63·11-s + 62.5i·12-s − 6.44i·13-s + 2.77·14-s − 11.2·16-s − 17i·17-s + 4.91i·18-s − 7.96·19-s + ⋯
L(s)  = 1  + 1.59i·2-s − 0.979i·3-s − 1.53·4-s + 1.56·6-s − 0.0332i·7-s − 0.854i·8-s + 0.0404·9-s − 0.236·11-s + 1.50i·12-s − 0.137i·13-s + 0.0530·14-s − 0.175·16-s − 0.242i·17-s + 0.0644i·18-s − 0.0962·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.205208747\)
\(L(\frac12)\) \(\approx\) \(1.205208747\)
\(L(\frac{5}{2})\) 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 \( 1 + 17iT \)
good2 \( 1 - 4.50iT - 8T^{2} \)
3 \( 1 + 5.08iT - 27T^{2} \)
7 \( 1 + 0.616iT - 343T^{2} \)
11 \( 1 + 8.63T + 1.33e3T^{2} \)
13 \( 1 + 6.44iT - 2.19e3T^{2} \)
19 \( 1 + 7.96T + 6.85e3T^{2} \)
23 \( 1 + 66.1iT - 1.21e4T^{2} \)
29 \( 1 + 219.T + 2.43e4T^{2} \)
31 \( 1 + 0.608T + 2.97e4T^{2} \)
37 \( 1 + 216. iT - 5.06e4T^{2} \)
41 \( 1 - 355.T + 6.89e4T^{2} \)
43 \( 1 + 209. iT - 7.95e4T^{2} \)
47 \( 1 + 324. iT - 1.03e5T^{2} \)
53 \( 1 + 189. iT - 1.48e5T^{2} \)
59 \( 1 - 257.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + 66.9iT - 3.00e5T^{2} \)
71 \( 1 + 131.T + 3.57e5T^{2} \)
73 \( 1 + 1.17e3iT - 3.89e5T^{2} \)
79 \( 1 + 707.T + 4.93e5T^{2} \)
83 \( 1 + 1.00e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 973. iT - 9.12e5T^{2} \)
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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

−10.61119395076131180551153305954, −9.368065077020431988812267424285, −8.479955372541029967243618131241, −7.55272734072311951936463383977, −7.12654491708700652488604538248, −6.16325254027213832010473283898, −5.34549033348503523438649078836, −4.10017889722380182068765176008, −2.18617497802484132945019063238, −0.41774442208704852881271318796, 1.33765747916344948802322575187, 2.69353123595327882118043111094, 3.79792306524857407690521996177, 4.47113534063383936214488745111, 5.63700364293141961916962762621, 7.23535792921403036498295674950, 8.599068401365761928760942179001, 9.573852678123670018658605710336, 9.929298231071283517029023404994, 10.97370376577925126066683034951

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