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Properties

Label 2-405-135.92-c1-0-15
Degree $2$
Conductor $405$
Sign $0.798 + 0.602i$
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  + (2.45 − 0.214i)2-s + (4.00 − 0.706i)4-s + (−1.21 − 1.87i)5-s + (1.02 − 0.719i)7-s + (4.92 − 1.32i)8-s + (−3.37 − 4.35i)10-s + (−0.955 − 2.62i)11-s + (−0.519 + 5.93i)13-s + (2.36 − 1.98i)14-s + (4.16 − 1.51i)16-s + (5.37 + 1.44i)17-s + (−1.75 − 1.01i)19-s + (−6.18 − 6.68i)20-s + (−2.90 − 6.23i)22-s + (−2.67 + 3.82i)23-s + ⋯
L(s)  = 1  + (1.73 − 0.151i)2-s + (2.00 − 0.353i)4-s + (−0.541 − 0.840i)5-s + (0.388 − 0.271i)7-s + (1.74 − 0.466i)8-s + (−1.06 − 1.37i)10-s + (−0.287 − 0.791i)11-s + (−0.143 + 1.64i)13-s + (0.632 − 0.530i)14-s + (1.04 − 0.378i)16-s + (1.30 + 0.349i)17-s + (−0.402 − 0.232i)19-s + (−1.38 − 1.49i)20-s + (−0.620 − 1.32i)22-s + (−0.558 + 0.798i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.798 + 0.602i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.798 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.16026 - 1.05830i\)
\(L(\frac12)\) \(\approx\) \(3.16026 - 1.05830i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 + 1.87i)T \)
good2 \( 1 + (-2.45 + 0.214i)T + (1.96 - 0.347i)T^{2} \)
7 \( 1 + (-1.02 + 0.719i)T + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (0.955 + 2.62i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.519 - 5.93i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (-5.37 - 1.44i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.75 + 1.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.67 - 3.82i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (2.13 + 1.79i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.860 - 4.87i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.354 + 1.32i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.207 - 0.247i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.79 + 8.13i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (-0.213 - 0.305i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (7.96 + 7.96i)T + 53iT^{2} \)
59 \( 1 + (2.97 + 1.08i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.0275 - 0.156i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.09 - 0.707i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (7.01 - 4.04i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.41 + 5.29i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.3 + 12.2i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.151 - 1.73i)T + (-81.7 + 14.4i)T^{2} \)
89 \( 1 + (6.44 - 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.21 + 1.49i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.56766203254107547639342737332, −10.77964831876277086760042892201, −9.363574091330146477694917291860, −8.194608355143703323193321530165, −7.19396219119213897236145822807, −6.01572701881970026700168212377, −5.12566554114430783453741795312, −4.23782713444668665386694387249, −3.44666684210775179436186675251, −1.73480675889668266889741055682, 2.48751199560862597927574099310, 3.35105866870697096113825759464, 4.48501486371627899762508645194, 5.46749121815125087647215079019, 6.31073677893459854367624477179, 7.51816353992334769194030274165, 7.972278849550520433553189262465, 9.949997294660862029200046482618, 10.75267166069739683783061521938, 11.64335555161647091629812082858

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