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Properties

Label 2-735-35.4-c1-0-10
Degree $2$
Conductor $735$
Sign $0.0736 - 0.997i$
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  + (0.167 − 0.0969i)2-s + (−0.866 − 0.5i)3-s + (−0.981 + 1.69i)4-s + (2.19 + 0.445i)5-s − 0.193·6-s + 0.768i·8-s + (0.499 + 0.866i)9-s + (0.411 − 0.137i)10-s + (−1 + 1.73i)11-s + (1.69 − 0.981i)12-s − 1.35i·13-s + (−1.67 − 1.48i)15-s + (−1.88 − 3.26i)16-s + (2.90 + 1.67i)17-s + (0.167 + 0.0969i)18-s + (2.67 + 4.63i)19-s + ⋯
L(s)  = 1  + (0.118 − 0.0685i)2-s + (−0.499 − 0.288i)3-s + (−0.490 + 0.849i)4-s + (0.979 + 0.199i)5-s − 0.0791·6-s + 0.271i·8-s + (0.166 + 0.288i)9-s + (0.130 − 0.0435i)10-s + (−0.301 + 0.522i)11-s + (0.490 − 0.283i)12-s − 0.374i·13-s + (−0.432 − 0.382i)15-s + (−0.471 − 0.817i)16-s + (0.703 + 0.406i)17-s + (0.0395 + 0.0228i)18-s + (0.613 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0736 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.0736 - 0.997i$
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: \((2,\ 735,\ (\ :1/2),\ 0.0736 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907627 + 0.843099i\)
\(L(\frac12)\) \(\approx\) \(0.907627 + 0.843099i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 0.445i)T \)
7 \( 1 \)
good2 \( 1 + (-0.167 + 0.0969i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.35iT - 13T^{2} \)
17 \( 1 + (-2.90 - 1.67i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.67 - 4.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + (2.28 - 3.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.671 - 0.387i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 + (-8.59 + 4.96i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.42 - 4.28i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.31 - 7.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.35 - 7.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.59 + 4.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (8.09 + 4.67i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.35 - 9.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.22iT - 83T^{2} \)
89 \( 1 + (-0.518 - 0.898i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.4iT - 97T^{2} \)
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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

−10.45265803676870719395384626251, −9.833023799274751320300383603948, −8.941269188467122935213178251788, −7.78168601912770893613350945190, −7.30337106518983036399384325045, −5.89555197535336081912497746734, −5.42256519813039620609067859381, −4.14234935909360390271802037329, −3.00013456199197483998078757076, −1.66073312672501879422302646716, 0.67875154723938648913771673678, 2.18666793887950695988208025001, 3.86975992356494657428738607309, 5.05195389554258283609083414513, 5.58171148433386644803582311831, 6.30369754613979380791446285953, 7.42094311338360886288594896543, 8.896235275701272264161262754719, 9.370758151663862177415798125219, 10.16199609312863869803924213819

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