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Properties

Label 2-735-7.6-c2-0-33
Degree $2$
Conductor $735$
Sign $-0.755 + 0.654i$
Analytic cond. $20.0272$
Root an. cond. $4.47518$
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  − 3.52·2-s − 1.73i·3-s + 8.39·4-s + 2.23i·5-s + 6.09i·6-s − 15.4·8-s − 2.99·9-s − 7.87i·10-s + 2.59·11-s − 14.5i·12-s + 11.5i·13-s + 3.87·15-s + 20.8·16-s + 23.2i·17-s + 10.5·18-s − 29.9i·19-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.577i·3-s + 2.09·4-s + 0.447i·5-s + 1.01i·6-s − 1.93·8-s − 0.333·9-s − 0.787i·10-s + 0.235·11-s − 1.21i·12-s + 0.890i·13-s + 0.258·15-s + 1.30·16-s + 1.36i·17-s + 0.586·18-s − 1.57i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(20.0272\)
Root analytic conductor: \(4.47518\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2746134984\)
\(L(\frac12)\) \(\approx\) \(0.2746134984\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 \)
good2 \( 1 + 3.52T + 4T^{2} \)
11 \( 1 - 2.59T + 121T^{2} \)
13 \( 1 - 11.5iT - 169T^{2} \)
17 \( 1 - 23.2iT - 289T^{2} \)
19 \( 1 + 29.9iT - 361T^{2} \)
23 \( 1 + 35.1T + 529T^{2} \)
29 \( 1 + 24.4T + 841T^{2} \)
31 \( 1 + 37.4iT - 961T^{2} \)
37 \( 1 - 25.7T + 1.36e3T^{2} \)
41 \( 1 - 3.71iT - 1.68e3T^{2} \)
43 \( 1 - 74.2T + 1.84e3T^{2} \)
47 \( 1 + 3.37iT - 2.20e3T^{2} \)
53 \( 1 + 40.0T + 2.80e3T^{2} \)
59 \( 1 + 49.3iT - 3.48e3T^{2} \)
61 \( 1 - 0.883iT - 3.72e3T^{2} \)
67 \( 1 + 65.0T + 4.48e3T^{2} \)
71 \( 1 - 86.0T + 5.04e3T^{2} \)
73 \( 1 + 61.5iT - 5.32e3T^{2} \)
79 \( 1 - 27.5T + 6.24e3T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 - 65.2iT - 7.92e3T^{2} \)
97 \( 1 - 42.2iT - 9.40e3T^{2} \)
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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

−9.593662686464598622830645908626, −9.094745526452580746757499609726, −8.087969198674185336157838681873, −7.51944226321493828242058309290, −6.58906389549101344897866193099, −6.02920446408254343573532730391, −4.09954209290173921331651602931, −2.48838476933225977109139078722, −1.67234613157416616217115100555, −0.18900218288418468605360274026, 1.15520348716861274274214709963, 2.52993938198524135802133659732, 3.88650901796273289683892113962, 5.34933484176220570502172687092, 6.26872257349318765204022446615, 7.56726995265413531978701517420, 8.023171360388413597295089213614, 8.968908521956468094882481386205, 9.614675447578294066171112303196, 10.22286927105519201726330927660

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