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Properties

Label 2-735-7.6-c2-0-15
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  + 0.673·2-s − 1.73i·3-s − 3.54·4-s − 2.23i·5-s − 1.16i·6-s − 5.08·8-s − 2.99·9-s − 1.50i·10-s − 0.0447·11-s + 6.14i·12-s + 23.0i·13-s − 3.87·15-s + 10.7·16-s − 9.42i·17-s − 2.02·18-s − 1.14i·19-s + ⋯
L(s)  = 1  + 0.336·2-s − 0.577i·3-s − 0.886·4-s − 0.447i·5-s − 0.194i·6-s − 0.635·8-s − 0.333·9-s − 0.150i·10-s − 0.00406·11-s + 0.511i·12-s + 1.76i·13-s − 0.258·15-s + 0.672·16-s − 0.554i·17-s − 0.112·18-s − 0.0602i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.209835974\)
\(L(\frac12)\) \(\approx\) \(1.209835974\)
\(L(2)\) 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 \( 1 - 0.673T + 4T^{2} \)
11 \( 1 + 0.0447T + 121T^{2} \)
13 \( 1 - 23.0iT - 169T^{2} \)
17 \( 1 + 9.42iT - 289T^{2} \)
19 \( 1 + 1.14iT - 361T^{2} \)
23 \( 1 + 44.2T + 529T^{2} \)
29 \( 1 - 53.0T + 841T^{2} \)
31 \( 1 - 22.5iT - 961T^{2} \)
37 \( 1 - 42.2T + 1.36e3T^{2} \)
41 \( 1 - 38.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.5T + 1.84e3T^{2} \)
47 \( 1 - 27.1iT - 2.20e3T^{2} \)
53 \( 1 - 18.9T + 2.80e3T^{2} \)
59 \( 1 + 4.86iT - 3.48e3T^{2} \)
61 \( 1 - 38.8iT - 3.72e3T^{2} \)
67 \( 1 + 7.00T + 4.48e3T^{2} \)
71 \( 1 + 46.8T + 5.04e3T^{2} \)
73 \( 1 - 83.5iT - 5.32e3T^{2} \)
79 \( 1 - 20.4T + 6.24e3T^{2} \)
83 \( 1 - 125. iT - 6.88e3T^{2} \)
89 \( 1 + 46.7iT - 7.92e3T^{2} \)
97 \( 1 - 3.11iT - 9.40e3T^{2} \)
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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

−10.05722140126156966474162007919, −9.320255909274795703204408040115, −8.602372658963096059537911634051, −7.79507977616643553554283030191, −6.62302452214445390384827844361, −5.84809177954041715993254212605, −4.64807523007360256280956696608, −4.09017102367875311135349515207, −2.55307509621148071699735988695, −1.09652000205247443199408834861, 0.46309672753214124894203958472, 2.66218972786278958352095586592, 3.68824882275149060352981894210, 4.47630061259233947279129795978, 5.62558613742328621040294673888, 6.13268786587278757014852225466, 7.79437988163388568249521044880, 8.285981332314709736789776979770, 9.351447988731673705751472007127, 10.25755326549546311940353699552

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