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Properties

Label 2-735-35.4-c1-0-27
Degree $2$
Conductor $735$
Sign $0.946 - 0.324i$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (−0.500 + 0.866i)4-s + (2.23 − 0.133i)5-s + 0.999·6-s + 3i·8-s + (0.499 + 0.866i)9-s + (1.86 − 1.23i)10-s + (3 − 5.19i)11-s + (−0.866 + 0.499i)12-s + 2i·13-s + (1.99 + i)15-s + (0.500 + 0.866i)16-s + (−3.46 − 2i)17-s + (0.866 + 0.499i)18-s + (3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (−0.250 + 0.433i)4-s + (0.998 − 0.0599i)5-s + 0.408·6-s + 1.06i·8-s + (0.166 + 0.288i)9-s + (0.590 − 0.389i)10-s + (0.904 − 1.56i)11-s + (−0.249 + 0.144i)12-s + 0.554i·13-s + (0.516 + 0.258i)15-s + (0.125 + 0.216i)16-s + (−0.840 − 0.485i)17-s + (0.204 + 0.117i)18-s + (0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.946 - 0.324i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.946 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71282 + 0.451791i\)
\(L(\frac12)\) \(\approx\) \(2.71282 + 0.451791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.8 + 8i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (5.19 + 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−10.52901955236667156895873122868, −9.312032967059044423277536687644, −8.926167347478795895950892617900, −8.100941365005593090051319245668, −6.77418104971359257305601355950, −5.78310921895299643788723749307, −4.88282234149568722260628041908, −3.74545392419460047277695905012, −3.03079065776706083995473783634, −1.72804471776758617385068342461, 1.37777713116025096595859470870, 2.56367300194036970690529985710, 4.09873221074532699234030132208, 4.88744909426449595568569494590, 5.95179792119099077758809161526, 6.71903661883687452442008352226, 7.40922127513720423469584755522, 8.875161507755185172629671792406, 9.527870602374225373956641393279, 10.00584846371950522782343507864

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