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Properties

Label 2-405-45.13-c2-0-36
Degree $2$
Conductor $405$
Sign $0.930 - 0.366i$
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  + (3.50 + 0.939i)2-s + (7.95 + 4.59i)4-s + (−1.14 − 4.86i)5-s + (6.20 + 1.66i)7-s + (13.3 + 13.3i)8-s + (0.573 − 18.1i)10-s + (−3.66 − 6.34i)11-s + (18.9 − 5.08i)13-s + (20.2 + 11.6i)14-s + (15.8 + 27.4i)16-s + (−20.9 + 20.9i)17-s − 0.814i·19-s + (13.2 − 43.9i)20-s + (−6.88 − 25.7i)22-s + (21.4 − 5.74i)23-s + ⋯
L(s)  = 1  + (1.75 + 0.469i)2-s + (1.98 + 1.14i)4-s + (−0.228 − 0.973i)5-s + (0.886 + 0.237i)7-s + (1.66 + 1.66i)8-s + (0.0573 − 1.81i)10-s + (−0.333 − 0.576i)11-s + (1.45 − 0.391i)13-s + (1.44 + 0.833i)14-s + (0.989 + 1.71i)16-s + (−1.23 + 1.23i)17-s − 0.0428i·19-s + (0.664 − 2.19i)20-s + (−0.313 − 1.16i)22-s + (0.931 − 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.930 - 0.366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.87972 + 0.925219i\)
\(L(\frac12)\) \(\approx\) \(4.87972 + 0.925219i\)
\(L(2)\) 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 \( 1 + (1.14 + 4.86i)T \)
good2 \( 1 + (-3.50 - 0.939i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (-6.20 - 1.66i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.66 + 6.34i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-18.9 + 5.08i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (20.9 - 20.9i)T - 289iT^{2} \)
19 \( 1 + 0.814iT - 361T^{2} \)
23 \( 1 + (-21.4 + 5.74i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (19.8 - 11.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (23.7 - 41.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.0 + 11.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-1.40 + 2.43i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-15.0 + 56.3i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (71.0 + 19.0i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-20.9 - 20.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (33.7 + 19.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-2.86 - 4.96i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.1 - 75.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 62.1T + 5.04e3T^{2} \)
73 \( 1 + (-8.47 - 8.47i)T + 5.32e3iT^{2} \)
79 \( 1 + (23.0 - 13.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (18.9 - 70.8i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 79.7iT - 7.92e3T^{2} \)
97 \( 1 + (84.2 + 22.5i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.17798348607776928083379863743, −10.92758226251774561293151710251, −8.693303046301213714484304746349, −8.352523249479287380535919321546, −7.06486480778370865576119213629, −5.90677846804999448879967276981, −5.26516772329866867838491208070, −4.31969101322750646012673091093, −3.41387429427424255691243346009, −1.70980817105435338621557650489, 1.82742300280271180677439305348, 2.95115037017431220751868744907, 4.07970046937756275796248497937, 4.83859429400806479738749981021, 6.05791581537566204237855528022, 6.88428333082142211070065493517, 7.80655589060049259743640662593, 9.436339975360436421456381083920, 10.79457775272875642720959684999, 11.28538259946923338593305220402

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