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Properties

Label 2-735-21.5-c1-0-17
Degree $2$
Conductor $735$
Sign $-0.817 - 0.575i$
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 + (0.627 + 1.61i)3-s + (1.12 + 1.95i)4-s + (−0.5 + 0.866i)5-s + (−0.543 + 3.53i)6-s + 0.527i·8-s + (−2.21 + 2.02i)9-s + (−1.78 + 1.03i)10-s + (−4.06 + 2.34i)11-s + (−2.44 + 3.04i)12-s + 0.638i·13-s + (−1.71 − 0.263i)15-s + (1.71 − 2.96i)16-s + (2.07 + 3.59i)17-s + (−6.04 + 1.33i)18-s + (0.776 + 0.448i)19-s + ⋯
L(s)  = 1  + (1.26 + 0.729i)2-s + (0.362 + 0.932i)3-s + (0.563 + 0.976i)4-s + (−0.223 + 0.387i)5-s + (−0.221 + 1.44i)6-s + 0.186i·8-s + (−0.737 + 0.675i)9-s + (−0.564 + 0.326i)10-s + (−1.22 + 0.707i)11-s + (−0.705 + 0.879i)12-s + 0.177i·13-s + (−0.442 − 0.0680i)15-s + (0.427 − 0.741i)16-s + (0.503 + 0.871i)17-s + (−1.42 + 0.315i)18-s + (0.178 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.817 - 0.575i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.817 - 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874972 + 2.76498i\)
\(L(\frac12)\) \(\approx\) \(0.874972 + 2.76498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.627 - 1.61i)T \)
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

−10.65389327480201572992644647484, −9.997707537431526259507093435968, −9.006670695710651586596014498189, −7.76674322864432484435392200537, −7.29601375614684349762880290502, −5.95059177630502989959152654797, −5.26957690570232267626130362919, −4.41083535832057156423852548391, −3.56284941499881468879245806407, −2.59499924407161380625965197197, 1.00039194940263104121922112778, 2.69269176717243667183364493081, 3.04669104709531701186880261049, 4.51312072579694594035276942943, 5.38088815164942216801778956328, 6.20920603136786199200119611992, 7.48551729436236092507175633276, 8.182772129901974762754071579468, 9.082774977087822666959574365377, 10.38716842791319415990263097783

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