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Properties

Label 2-735-7.6-c2-0-26
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  − 1.67·2-s + 1.73i·3-s − 1.19·4-s + 2.23i·5-s − 2.89i·6-s + 8.70·8-s − 2.99·9-s − 3.74i·10-s − 13.8·11-s − 2.07i·12-s + 6.12i·13-s − 3.87·15-s − 9.76·16-s + 2.47i·17-s + 5.02·18-s − 27.9i·19-s + ⋯
L(s)  = 1  − 0.836·2-s + 0.577i·3-s − 0.299·4-s + 0.447i·5-s − 0.483i·6-s + 1.08·8-s − 0.333·9-s − 0.374i·10-s − 1.25·11-s − 0.173i·12-s + 0.470i·13-s − 0.258·15-s − 0.610·16-s + 0.145i·17-s + 0.278·18-s − 1.47i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.5310210567\)
\(L(\frac12)\) \(\approx\) \(0.5310210567\)
\(L(2)\) 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 \( 1 + 1.67T + 4T^{2} \)
11 \( 1 + 13.8T + 121T^{2} \)
13 \( 1 - 6.12iT - 169T^{2} \)
17 \( 1 - 2.47iT - 289T^{2} \)
19 \( 1 + 27.9iT - 361T^{2} \)
23 \( 1 - 13.2T + 529T^{2} \)
29 \( 1 + 27.6T + 841T^{2} \)
31 \( 1 - 18.7iT - 961T^{2} \)
37 \( 1 + 41.0T + 1.36e3T^{2} \)
41 \( 1 + 22.5iT - 1.68e3T^{2} \)
43 \( 1 - 7.60T + 1.84e3T^{2} \)
47 \( 1 + 13.7iT - 2.20e3T^{2} \)
53 \( 1 - 92.5T + 2.80e3T^{2} \)
59 \( 1 - 71.0iT - 3.48e3T^{2} \)
61 \( 1 + 115. iT - 3.72e3T^{2} \)
67 \( 1 + 11.4T + 4.48e3T^{2} \)
71 \( 1 - 99.4T + 5.04e3T^{2} \)
73 \( 1 + 104. iT - 5.32e3T^{2} \)
79 \( 1 - 128.T + 6.24e3T^{2} \)
83 \( 1 + 30.3iT - 6.88e3T^{2} \)
89 \( 1 - 108. iT - 7.92e3T^{2} \)
97 \( 1 + 153. iT - 9.40e3T^{2} \)
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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

−10.07466503214298036989885604346, −9.181247184436932588957358097783, −8.631949110516995782611890129293, −7.59981785967708552847127175013, −6.87625727488255032149025776986, −5.39474473287543713377832270674, −4.70492929895718795805754288373, −3.48353551991119235878431488168, −2.19503728623227483054022245458, −0.32932693602756147693850814332, 0.910274393717926327208867959433, 2.18585136264645906649150304199, 3.71634751372423090766056611719, 5.05066593324625947262877571180, 5.74051045345546400381000358480, 7.18415647396576101448671749527, 7.911714127583387607283841865624, 8.408633113763735748494050895969, 9.362476433267989788022993772630, 10.19388831585700196873263407209

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