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Properties

Label 2-405-5.4-c3-0-62
Degree $2$
Conductor $405$
Sign $-0.323 - 0.946i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
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  − 3.99i·2-s − 7.98·4-s + (−3.61 − 10.5i)5-s − 7.79i·7-s − 0.0620i·8-s + (−42.3 + 14.4i)10-s + 32.7·11-s − 65.9i·13-s − 31.1·14-s − 64.1·16-s − 15.5i·17-s − 51.6·19-s + (28.8 + 84.4i)20-s − 130. i·22-s − 49.6i·23-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.998·4-s + (−0.323 − 0.946i)5-s − 0.421i·7-s − 0.00274i·8-s + (−1.33 + 0.456i)10-s + 0.896·11-s − 1.40i·13-s − 0.595·14-s − 1.00·16-s − 0.222i·17-s − 0.623·19-s + (0.322 + 0.944i)20-s − 1.26i·22-s − 0.450i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.311842276\)
\(L(\frac12)\) \(\approx\) \(1.311842276\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (3.61 + 10.5i)T \)
good2 \( 1 + 3.99iT - 8T^{2} \)
7 \( 1 + 7.79iT - 343T^{2} \)
11 \( 1 - 32.7T + 1.33e3T^{2} \)
13 \( 1 + 65.9iT - 2.19e3T^{2} \)
17 \( 1 + 15.5iT - 4.91e3T^{2} \)
19 \( 1 + 51.6T + 6.85e3T^{2} \)
23 \( 1 + 49.6iT - 1.21e4T^{2} \)
29 \( 1 - 282.T + 2.43e4T^{2} \)
31 \( 1 + 147.T + 2.97e4T^{2} \)
37 \( 1 + 122. iT - 5.06e4T^{2} \)
41 \( 1 + 109.T + 6.89e4T^{2} \)
43 \( 1 - 191. iT - 7.95e4T^{2} \)
47 \( 1 - 414. iT - 1.03e5T^{2} \)
53 \( 1 - 236. iT - 1.48e5T^{2} \)
59 \( 1 + 784.T + 2.05e5T^{2} \)
61 \( 1 + 535.T + 2.26e5T^{2} \)
67 \( 1 + 121. iT - 3.00e5T^{2} \)
71 \( 1 - 590.T + 3.57e5T^{2} \)
73 \( 1 - 986. iT - 3.89e5T^{2} \)
79 \( 1 - 607.T + 4.93e5T^{2} \)
83 \( 1 + 1.29e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 - 610. iT - 9.12e5T^{2} \)
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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

−10.43363385519694239933257580834, −9.457051466850229004130440804064, −8.676359808014508913423175844076, −7.63454449034131721839255562698, −6.27580577195082117374025687570, −4.82312726763114911224464773780, −3.99906930749119362732476346892, −2.90567815590252675384135628049, −1.37428827520788215461600059400, −0.46019798840831088073270643596, 2.11221747861135453171227275832, 3.75175313167767002382161146576, 4.86917246501143723300525642587, 6.29991282456373578716902377741, 6.59590980292324639356247872692, 7.52170721405154637893201290082, 8.570074064442885733686779809812, 9.262675287838411916095414418396, 10.53556680428266796988655226296, 11.55907724581170739790351026250

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