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Properties

Label 2-405-135.68-c1-0-10
Degree $2$
Conductor $405$
Sign $0.742 - 0.669i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (−1.15 + 1.65i)2-s + (−0.704 − 1.93i)4-s + (2.13 − 0.650i)5-s + (1.32 − 0.618i)7-s + (0.115 + 0.0310i)8-s + (−1.39 + 4.28i)10-s + (−1.86 − 2.21i)11-s + (5.02 − 3.51i)13-s + (−0.512 + 2.90i)14-s + (2.97 − 2.49i)16-s + (3.10 − 0.833i)17-s + (−3.51 + 2.03i)19-s + (−2.76 − 3.68i)20-s + (5.80 − 0.508i)22-s + (2.14 − 4.58i)23-s + ⋯
L(s)  = 1  + (−0.817 + 1.16i)2-s + (−0.352 − 0.967i)4-s + (0.956 − 0.290i)5-s + (0.501 − 0.233i)7-s + (0.0409 + 0.0109i)8-s + (−0.442 + 1.35i)10-s + (−0.560 − 0.668i)11-s + (1.39 − 0.975i)13-s + (−0.136 + 0.776i)14-s + (0.742 − 0.623i)16-s + (0.753 − 0.202i)17-s + (−0.806 + 0.465i)19-s + (−0.618 − 0.823i)20-s + (1.23 − 0.108i)22-s + (0.446 − 0.957i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.742 - 0.669i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02172 + 0.392572i\)
\(L(\frac12)\) \(\approx\) \(1.02172 + 0.392572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.13 + 0.650i)T \)
good2 \( 1 + (1.15 - 1.65i)T + (-0.684 - 1.87i)T^{2} \)
7 \( 1 + (-1.32 + 0.618i)T + (4.49 - 5.36i)T^{2} \)
11 \( 1 + (1.86 + 2.21i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-5.02 + 3.51i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (-3.10 + 0.833i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.51 - 2.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 4.58i)T + (-14.7 - 17.6i)T^{2} \)
29 \( 1 + (-0.368 - 2.08i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.20 - 1.16i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.04 - 11.3i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.839 + 0.147i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.732 + 0.0641i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (-1.21 - 2.61i)T + (-30.2 + 36.0i)T^{2} \)
53 \( 1 + (0.0757 - 0.0757i)T - 53iT^{2} \)
59 \( 1 + (2.89 + 2.43i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (4.21 + 1.53i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.55 + 3.65i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-7.84 - 4.53i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.15 + 8.04i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.44 - 0.254i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-13.8 - 9.69i)T + (28.3 + 77.9i)T^{2} \)
89 \( 1 + (-3.05 - 5.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.41 + 16.1i)T + (-95.5 - 16.8i)T^{2} \)
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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

−10.88944753832517065393752783828, −10.33757806027582558488730472760, −9.263466152822808134013712802380, −8.328504472221914140292081850873, −8.007639748272548792486287892467, −6.55726880253869585846336019984, −5.91416482383304748424466274620, −5.02824133745139073757290248191, −3.13814796715589547324074807096, −1.10012823873658733644175631661, 1.52972795913473482889638312019, 2.36606427194321780090773369896, 3.78572125272400722528233451658, 5.37787868378768635059259880617, 6.37845640449982531657451656711, 7.72418874612335481885459598024, 8.893501772416444944662117861680, 9.365027699417441185841512595843, 10.37778244925193023390715122713, 10.96731489417380544178773095831

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