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Properties

Label 2-735-105.32-c1-0-43
Degree $2$
Conductor $735$
Sign $0.113 - 0.993i$
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  + (2.10 + 0.565i)2-s + (1.49 + 0.871i)3-s + (2.39 + 1.38i)4-s + (−1.79 + 1.32i)5-s + (2.66 + 2.68i)6-s + (1.18 + 1.18i)8-s + (1.48 + 2.60i)9-s + (−4.54 + 1.78i)10-s + (2.93 + 1.69i)11-s + (2.38 + 4.16i)12-s + (0.206 − 0.206i)13-s + (−3.84 + 0.423i)15-s + (−0.935 − 1.62i)16-s + (0.0612 + 0.228i)17-s + (1.65 + 6.34i)18-s + (−4.60 + 2.65i)19-s + ⋯
L(s)  = 1  + (1.49 + 0.399i)2-s + (0.864 + 0.503i)3-s + (1.19 + 0.692i)4-s + (−0.804 + 0.594i)5-s + (1.08 + 1.09i)6-s + (0.419 + 0.419i)8-s + (0.493 + 0.869i)9-s + (−1.43 + 0.565i)10-s + (0.883 + 0.510i)11-s + (0.687 + 1.20i)12-s + (0.0573 − 0.0573i)13-s + (−0.994 + 0.109i)15-s + (−0.233 − 0.405i)16-s + (0.0148 + 0.0554i)17-s + (0.388 + 1.49i)18-s + (−1.05 + 0.609i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.94982 + 2.63147i\)
\(L(\frac12)\) \(\approx\) \(2.94982 + 2.63147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.49 - 0.871i)T \)
5 \( 1 + (1.79 - 1.32i)T \)
7 \( 1 \)
good2 \( 1 + (-2.10 - 0.565i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.206 + 0.206i)T - 13iT^{2} \)
17 \( 1 + (-0.0612 - 0.228i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.60 - 2.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 6.93i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.84T + 29T^{2} \)
31 \( 1 + (-4.55 + 7.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.92 - 7.19i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 - 3.76i)T - 43iT^{2} \)
47 \( 1 + (-4.87 - 1.30i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.85 + 1.30i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.15 + 8.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.67 - 2.32i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (3.15 + 11.7i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.91 - 5.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.88 - 3.88i)T + 83iT^{2} \)
89 \( 1 + (1.00 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.26 + 2.26i)T + 97iT^{2} \)
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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

−10.65929484188886255373553867112, −9.825697109077552009963824190365, −8.643429605180712287033090401103, −7.86623348261162227164721208706, −6.83455159631891692807112047774, −6.24604881016122519592023744632, −4.65978302922952892341454740982, −4.25708733567409010842282361719, −3.37673700392347923257095413053, −2.40864525940348657869293678432, 1.37561098386976174048903174646, 2.80461559598461018368302524775, 3.72704415416012177107760938108, 4.33757627342098882310008064866, 5.48470741623595358055605044485, 6.60754820592722633021640378464, 7.37737460214633396795346813147, 8.776630961966879122649537352831, 8.834600086751004755515996158176, 10.45653140442898288938170229566

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