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Properties

Label 2-735-49.43-c1-0-17
Degree $2$
Conductor $735$
Sign $0.950 + 0.311i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (−0.902 + 1.13i)2-s + (−0.900 − 0.433i)3-s + (−0.0206 − 0.0906i)4-s + (0.900 + 0.433i)5-s + (1.30 − 0.627i)6-s + (−2.26 − 1.36i)7-s + (−2.48 − 1.19i)8-s + (0.623 + 0.781i)9-s + (−1.30 + 0.627i)10-s + (−0.0398 + 0.0499i)11-s + (−0.0206 + 0.0906i)12-s + (−1.62 + 2.03i)13-s + (3.58 − 1.32i)14-s + (−0.623 − 0.781i)15-s + (3.76 − 1.81i)16-s + (0.785 − 3.44i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.799i)2-s + (−0.520 − 0.250i)3-s + (−0.0103 − 0.0453i)4-s + (0.402 + 0.194i)5-s + (0.532 − 0.256i)6-s + (−0.856 − 0.516i)7-s + (−0.878 − 0.423i)8-s + (0.207 + 0.260i)9-s + (−0.412 + 0.198i)10-s + (−0.0120 + 0.0150i)11-s + (−0.00597 + 0.0261i)12-s + (−0.449 + 0.563i)13-s + (0.959 − 0.355i)14-s + (−0.160 − 0.201i)15-s + (0.940 − 0.453i)16-s + (0.190 − 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.950 + 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.636344 - 0.101484i\)
\(L(\frac12)\) \(\approx\) \(0.636344 - 0.101484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (2.26 + 1.36i)T \)
good2 \( 1 + (0.902 - 1.13i)T + (-0.445 - 1.94i)T^{2} \)
11 \( 1 + (0.0398 - 0.0499i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.62 - 2.03i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.785 + 3.44i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 + (1.16 + 5.10i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-0.633 + 2.77i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + (-0.337 + 1.47i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (7.44 + 3.58i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-8.68 + 4.18i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.81 + 4.77i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.62 - 7.11i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-8.51 + 4.09i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-2.14 + 9.41i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (1.88 + 8.25i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-7.87 - 9.88i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 1.99T + 79T^{2} \)
83 \( 1 + (7.95 + 9.97i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.212 + 0.266i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−9.977560492257502772117211102341, −9.560409841259649877669845203120, −8.579693231240973566577257001146, −7.38763663365846301738999727676, −7.01199555488108122469356299053, −6.22424120494025730560360692541, −5.27634528800797125711187635406, −3.83366974196997035464252095937, −2.57143975986880596561597337413, −0.49572653045170976669230355469, 1.17728602730645580681108715429, 2.58325888532827934047587879508, 3.61782996303579194164777454045, 5.34652098577351535943859054360, 5.74511294575147733295429819498, 6.81030706192893658048110324020, 8.120298325573259622782291901636, 9.152350410305993983362718649686, 9.750713431532678641878541096286, 10.23314979025942924824376952128

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