Properties

Label 2-735-21.17-c1-0-5
Degree $2$
Conductor $735$
Sign $-0.953 + 0.302i$
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  + (−1.78 + 1.03i)2-s + (−1.08 + 1.35i)3-s + (1.12 − 1.95i)4-s + (0.5 + 0.866i)5-s + (0.543 − 3.53i)6-s + 0.527i·8-s + (−0.649 − 2.92i)9-s + (−1.78 − 1.03i)10-s + (4.06 + 2.34i)11-s + (1.41 + 3.64i)12-s − 0.638i·13-s + (−1.71 − 0.263i)15-s + (1.71 + 2.96i)16-s + (−2.07 + 3.59i)17-s + (4.18 + 4.56i)18-s + (0.776 − 0.448i)19-s + ⋯
L(s)  = 1  + (−1.26 + 0.729i)2-s + (−0.625 + 0.779i)3-s + (0.563 − 0.976i)4-s + (0.223 + 0.387i)5-s + (0.221 − 1.44i)6-s + 0.186i·8-s + (−0.216 − 0.976i)9-s + (−0.564 − 0.326i)10-s + (1.22 + 0.707i)11-s + (0.408 + 1.05i)12-s − 0.177i·13-s + (−0.442 − 0.0680i)15-s + (0.427 + 0.741i)16-s + (−0.503 + 0.871i)17-s + (0.985 + 1.07i)18-s + (0.178 − 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.953 + 0.302i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.953 + 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0709588 - 0.457888i\)
\(L(\frac12)\) \(\approx\) \(0.0709588 - 0.457888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.35i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.78 - 1.03i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-4.06 - 2.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.638iT - 13T^{2} \)
17 \( 1 + (2.07 - 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.776 + 0.448i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.89 - 3.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.14iT - 29T^{2} \)
31 \( 1 + (-2.02 - 1.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.69 - 9.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (3.40 + 5.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.96 + 1.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.254 - 0.440i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.48 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.22iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 7.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 + (6.90 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−10.34494798973475250325185245681, −9.975744228682386487986280032261, −9.207861975081687470925185859747, −8.462950181729917573035599157715, −7.35929026285637804711967063539, −6.46423064672026562299640384599, −5.99792244267821973731660472157, −4.56048038747747351703932390388, −3.55901376781911530314397102479, −1.49432980129507471105714550877, 0.42831850841198571723677883419, 1.52890066081788449923778387740, 2.61079952171386686701400025288, 4.32463207109042121002242461292, 5.66991431711035109860774654227, 6.48769961257876671150429076843, 7.55127037540905949877712103360, 8.365511457692565457717786605619, 9.130316019135914024974918719260, 9.841848147971877358561816238747

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