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Properties

Label 2-735-105.53-c1-0-54
Degree $2$
Conductor $735$
Sign $0.634 + 0.772i$
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  + (−0.127 + 0.474i)2-s + (1.01 − 1.40i)3-s + (1.52 + 0.879i)4-s + (1.06 − 1.96i)5-s + (0.536 + 0.659i)6-s + (−1.30 + 1.30i)8-s + (−0.942 − 2.84i)9-s + (0.795 + 0.755i)10-s + (−2.31 − 1.33i)11-s + (2.78 − 1.24i)12-s + (−2.14 − 2.14i)13-s + (−1.67 − 3.49i)15-s + (1.30 + 2.26i)16-s + (4.46 − 1.19i)17-s + (1.46 − 0.0850i)18-s + (4.54 − 2.62i)19-s + ⋯
L(s)  = 1  + (−0.0898 + 0.335i)2-s + (0.585 − 0.810i)3-s + (0.761 + 0.439i)4-s + (0.477 − 0.878i)5-s + (0.219 + 0.269i)6-s + (−0.461 + 0.461i)8-s + (−0.314 − 0.949i)9-s + (0.251 + 0.238i)10-s + (−0.697 − 0.402i)11-s + (0.802 − 0.359i)12-s + (−0.596 − 0.596i)13-s + (−0.432 − 0.901i)15-s + (0.326 + 0.565i)16-s + (1.08 − 0.290i)17-s + (0.346 − 0.0200i)18-s + (1.04 − 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95005 - 0.921797i\)
\(L(\frac12)\) \(\approx\) \(1.95005 - 0.921797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 + 1.40i)T \)
5 \( 1 + (-1.06 + 1.96i)T \)
7 \( 1 \)
good2 \( 1 + (0.127 - 0.474i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (2.31 + 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.14 + 2.14i)T + 13iT^{2} \)
17 \( 1 + (-4.46 + 1.19i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.54 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.48 - 0.932i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 0.784i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \)
47 \( 1 + (-2.80 + 10.4i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.62 - 6.05i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.0797 + 0.138i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.98 + 7.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (-5.68 + 1.52i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.37 - 1.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 - 4.03i)T - 83iT^{2} \)
89 \( 1 + (-1.97 - 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.86 + 1.86i)T - 97iT^{2} \)
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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

−10.06274659067814982741206006732, −9.193061023666194004009439440918, −8.326168246547102535534585613545, −7.67391386670225700778352660812, −7.03025922546409402020432800675, −5.83079931730216513964260270274, −5.17963434847344757896378312232, −3.29990837503744961680720838569, −2.54677291321939403475484965071, −1.12340106852992814814673667040, 1.92891071656177348597322679029, 2.78660604664118185769292112148, 3.69198916784241270114334696109, 5.21150349289698359688245570723, 5.91486252213465238187755589920, 7.24597343940012599016794910826, 7.69296147221233731572978246746, 9.262197623983407637855767710876, 9.786983643303566521825043644739, 10.41072976098641782070799649906

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