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Properties

Label 2-405-45.13-c2-0-38
Degree $2$
Conductor $405$
Sign $0.793 - 0.608i$
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  + (3.68 + 0.986i)2-s + (9.10 + 5.25i)4-s + (4.36 − 2.43i)5-s + (1.55 + 0.416i)7-s + (17.5 + 17.5i)8-s + (18.4 − 4.67i)10-s + (−4.96 − 8.59i)11-s + (−19.3 + 5.19i)13-s + (5.31 + 3.06i)14-s + (26.2 + 45.4i)16-s + (6.77 − 6.77i)17-s − 4.72i·19-s + (52.5 + 0.730i)20-s + (−9.78 − 36.5i)22-s + (−13.7 + 3.68i)23-s + ⋯
L(s)  = 1  + (1.84 + 0.493i)2-s + (2.27 + 1.31i)4-s + (0.872 − 0.487i)5-s + (0.222 + 0.0595i)7-s + (2.19 + 2.19i)8-s + (1.84 − 0.467i)10-s + (−0.451 − 0.781i)11-s + (−1.49 + 0.399i)13-s + (0.379 + 0.219i)14-s + (1.64 + 2.84i)16-s + (0.398 − 0.398i)17-s − 0.248i·19-s + (2.62 + 0.0365i)20-s + (−0.444 − 1.66i)22-s + (−0.597 + 0.160i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.793 - 0.608i$
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: \((2,\ 405,\ (\ :1),\ 0.793 - 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.28680 + 1.79391i\)
\(L(\frac12)\) \(\approx\) \(5.28680 + 1.79391i\)
\(L(2)\) 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 \( 1 + (-4.36 + 2.43i)T \)
good2 \( 1 + (-3.68 - 0.986i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (-1.55 - 0.416i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (4.96 + 8.59i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (19.3 - 5.19i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-6.77 + 6.77i)T - 289iT^{2} \)
19 \( 1 + 4.72iT - 361T^{2} \)
23 \( 1 + (13.7 - 3.68i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (17.4 - 10.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (17.4 - 30.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-29.5 + 29.5i)T - 1.36e3iT^{2} \)
41 \( 1 + (18.8 - 32.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (5.61 - 20.9i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-21.0 - 5.64i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-44.2 - 44.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (-6.89 - 3.97i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.08 + 8.80i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.2 + 56.9i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 100.T + 5.04e3T^{2} \)
73 \( 1 + (91.7 + 91.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-7.24 + 4.18i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-11.2 + 41.9i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 87.6iT - 7.92e3T^{2} \)
97 \( 1 + (-28.0 - 7.50i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.55705447905300623036014897719, −10.51309310676162890583536583323, −9.298008391602447360991008709822, −7.965753193825610797542738749517, −7.07869250678243526953588280872, −6.00680641001079793469836099084, −5.25705258046842506575501334844, −4.58564088478884683609709929942, −3.13320275361190154914397679358, −2.06718098049357384337375000513, 1.94191741475439624207530036818, 2.67032153155940708479975943896, 4.01124356368942081385680452602, 5.12921856325542944845241015994, 5.74198853775311295872448897798, 6.85040594275268099174300312676, 7.67594997030590471552955019909, 9.835257430718465587940895895823, 10.15814777265368720405835501316, 11.17211782525054504561272179372

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