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Properties

Label 2-405-135.83-c1-0-1
Degree $2$
Conductor $405$
Sign $-0.766 - 0.642i$
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  + (−0.0928 + 0.0649i)2-s + (−0.679 + 1.86i)4-s + (−0.584 + 2.15i)5-s + (0.344 − 0.739i)7-s + (−0.116 − 0.436i)8-s + (−0.0860 − 0.238i)10-s + (0.792 − 0.944i)11-s + (−2.08 + 2.97i)13-s + (0.0160 + 0.0910i)14-s + (−3.00 − 2.52i)16-s + (−1.65 + 6.18i)17-s + (−2.78 − 1.60i)19-s + (−3.63 − 2.55i)20-s + (−0.0121 + 0.139i)22-s + (−0.471 + 0.219i)23-s + ⋯
L(s)  = 1  + (−0.0656 + 0.0459i)2-s + (−0.339 + 0.933i)4-s + (−0.261 + 0.965i)5-s + (0.130 − 0.279i)7-s + (−0.0413 − 0.154i)8-s + (−0.0272 − 0.0753i)10-s + (0.238 − 0.284i)11-s + (−0.577 + 0.825i)13-s + (0.00429 + 0.0243i)14-s + (−0.751 − 0.630i)16-s + (−0.402 + 1.50i)17-s + (−0.638 − 0.368i)19-s + (−0.812 − 0.571i)20-s + (−0.00259 + 0.0296i)22-s + (−0.0982 + 0.0458i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293245 + 0.806813i\)
\(L(\frac12)\) \(\approx\) \(0.293245 + 0.806813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.584 - 2.15i)T \)
good2 \( 1 + (0.0928 - 0.0649i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (-0.344 + 0.739i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-0.792 + 0.944i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (2.08 - 2.97i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.65 - 6.18i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.78 + 1.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.471 - 0.219i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.754 - 4.28i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (6.93 + 2.52i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-8.35 - 2.23i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.94 + 1.57i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.490 - 5.60i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (2.14 + 1.00i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (4.55 - 4.55i)T - 53iT^{2} \)
59 \( 1 + (-10.5 + 8.86i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-3.01 + 1.09i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-9.77 - 6.84i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (4.14 - 2.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.46 - 1.46i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.54 - 1.68i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.76 + 6.80i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-0.689 + 1.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.18 - 0.540i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.39868514943392146592381915627, −10.95456850706901811612225100100, −9.731205029843848019254323649364, −8.782873425912953358421763156956, −7.84588080385667320858040264394, −7.05868832150700938396815127504, −6.15797873145529914520846011527, −4.40050586573891627900839110940, −3.70160913299168623873546761434, −2.35930057351484156081364663766, 0.56502198693212961829104857090, 2.22441426728625587152292429819, 4.18244648983118945479674650694, 5.08955746824318917085138769171, 5.82409727552272898389387002746, 7.22206831714622144480536931298, 8.298860234271794642725789025253, 9.254345372718449755641583044845, 9.759680086060060061441892524006, 10.90505602959910623725755170731

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