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Properties

Label 2-405-45.7-c2-0-12
Degree $2$
Conductor $405$
Sign $-0.407 - 0.913i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.03 + 0.814i)2-s + (5.10 − 2.94i)4-s + (−2.32 + 4.42i)5-s + (1.98 − 0.530i)7-s + (−4.22 + 4.22i)8-s + (3.44 − 15.3i)10-s + (−1.67 + 2.89i)11-s + (14.2 + 3.82i)13-s + (−5.58 + 3.22i)14-s + (−2.39 + 4.15i)16-s + (2.65 + 2.65i)17-s − 20.6i·19-s + (1.20 + 29.4i)20-s + (2.72 − 10.1i)22-s + (22.4 + 6.02i)23-s + ⋯
L(s)  = 1  + (−1.51 + 0.407i)2-s + (1.27 − 0.737i)4-s + (−0.464 + 0.885i)5-s + (0.282 − 0.0757i)7-s + (−0.528 + 0.528i)8-s + (0.344 − 1.53i)10-s + (−0.152 + 0.263i)11-s + (1.09 + 0.294i)13-s + (−0.398 + 0.230i)14-s + (−0.149 + 0.259i)16-s + (0.155 + 0.155i)17-s − 1.08i·19-s + (0.0600 + 1.47i)20-s + (0.123 − 0.462i)22-s + (0.976 + 0.261i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.407 - 0.913i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.407 - 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.355846 + 0.548489i\)
\(L(\frac12)\) \(\approx\) \(0.355846 + 0.548489i\)
\(L(2)\) 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 \( 1 + (2.32 - 4.42i)T \)
good2 \( 1 + (3.03 - 0.814i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (-1.98 + 0.530i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (1.67 - 2.89i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.2 - 3.82i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (-2.65 - 2.65i)T + 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (-22.4 - 6.02i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-0.739 - 0.426i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-9.34 - 16.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-38.0 - 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (14.3 + 24.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-8.23 - 30.7i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-26.9 + 7.22i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (28.6 - 28.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (96.9 - 55.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (47.0 - 81.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (20.0 - 74.9i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 - 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (21.2 + 12.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-7.74 - 28.8i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (19.9 - 5.34i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−10.99623213691624402412672701974, −10.42154221689851595196413752930, −9.357704995250140917402215484412, −8.571479379526026450240252089406, −7.68720614155482543046242980907, −6.96257054789002082255797121890, −6.14021028991477625487500344498, −4.41245612638247701193988005671, −2.88654416729815232689113796187, −1.21589841370422207860629340324, 0.56249052828290058000089644482, 1.68817904993965251200306811040, 3.38898048487539329307065931797, 4.83827883354978886377932303615, 6.13339324183736543403213682741, 7.64367126589455280914562422413, 8.151730987285044503649135648200, 8.919751928289986932814290625506, 9.659589165322584855329204779079, 10.77564760482505888345612998328

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