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Properties

Label 2-735-21.17-c1-0-14
Degree $2$
Conductor $735$
Sign $-0.683 - 0.729i$
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  + (0.686 − 0.396i)2-s + (0.5 + 1.65i)3-s + (−0.686 + 1.18i)4-s + (0.5 + 0.866i)5-s + (1 + 0.939i)6-s + 2.67i·8-s + (−2.5 + 1.65i)9-s + (0.686 + 0.396i)10-s + (−2.18 − 1.26i)11-s + (−2.31 − 0.543i)12-s + 4.10i·13-s + (−1.18 + 1.26i)15-s + (−0.313 − 0.543i)16-s + (2.18 − 3.78i)17-s + (−1.05 + 2.12i)18-s + (3 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.485 − 0.280i)2-s + (0.288 + 0.957i)3-s + (−0.343 + 0.594i)4-s + (0.223 + 0.387i)5-s + (0.408 + 0.383i)6-s + 0.944i·8-s + (−0.833 + 0.552i)9-s + (0.216 + 0.125i)10-s + (−0.659 − 0.380i)11-s + (−0.667 − 0.156i)12-s + 1.13i·13-s + (−0.306 + 0.325i)15-s + (−0.0784 − 0.135i)16-s + (0.530 − 0.918i)17-s + (−0.249 + 0.501i)18-s + (0.688 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.683 - 0.729i$
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: \((2,\ 735,\ (\ :1/2),\ -0.683 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652512 + 1.50597i\)
\(L(\frac12)\) \(\approx\) \(0.652512 + 1.50597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 1.65i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.686 + 0.396i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.18 + 1.26i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.37 - 4.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.939iT - 29T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.37 - 5.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 + (0.813 + 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.62 + 0.939i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.37 - 7.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 - 4.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.294iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 + (-7.37 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−10.76186439817755617188603540036, −9.715981054114289735405287981660, −9.224013144990504241904992142732, −8.149632697978055610732198399363, −7.45880240129662838638789044583, −5.96508734323272073282834499495, −5.04824653657741993670851779322, −4.20625042380696958928242831219, −3.24676274462506374744076703048, −2.43317660087472294055056114526, 0.69564055562784098124881469655, 2.07773386110732105452851560189, 3.50551171614781702072537471779, 4.76801779243418368096395498179, 5.87901866868439699910585852474, 6.13227609637005919296036038386, 7.62573976531341563365935116899, 8.045966750776395923678122912824, 9.193291619346704615743679203114, 10.04858824396397860248359667547

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