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Properties

Label 2-735-105.2-c1-0-11
Degree $2$
Conductor $735$
Sign $0.580 + 0.814i$
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  + (0.658 + 2.45i)2-s + (−1.19 + 1.25i)3-s + (−3.87 + 2.23i)4-s + (0.108 + 2.23i)5-s + (−3.87 − 2.10i)6-s + (−4.45 − 4.45i)8-s + (−0.154 − 2.99i)9-s + (−5.41 + 1.73i)10-s + (1.35 − 0.784i)11-s + (1.81 − 7.53i)12-s + (−2.21 + 2.21i)13-s + (−2.93 − 2.52i)15-s + (3.54 − 6.14i)16-s + (−4.92 − 1.32i)17-s + (7.26 − 2.35i)18-s + (−1.45 − 0.840i)19-s + ⋯
L(s)  = 1  + (0.465 + 1.73i)2-s + (−0.688 + 0.725i)3-s + (−1.93 + 1.11i)4-s + (0.0485 + 0.998i)5-s + (−1.58 − 0.859i)6-s + (−1.57 − 1.57i)8-s + (−0.0514 − 0.998i)9-s + (−1.71 + 0.549i)10-s + (0.409 − 0.236i)11-s + (0.523 − 2.17i)12-s + (−0.615 + 0.615i)13-s + (−0.757 − 0.652i)15-s + (0.886 − 1.53i)16-s + (−1.19 − 0.320i)17-s + (1.71 − 0.554i)18-s + (−0.333 − 0.192i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676972 - 0.348675i\)
\(L(\frac12)\) \(\approx\) \(0.676972 - 0.348675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.19 - 1.25i)T \)
5 \( 1 + (-0.108 - 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (-0.658 - 2.45i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.35 + 0.784i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.21 - 2.21i)T - 13iT^{2} \)
17 \( 1 + (4.92 + 1.32i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.45 + 0.840i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.36 + 0.364i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + (-1.37 - 2.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.601 - 0.161i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 - 5.47i)T - 43iT^{2} \)
47 \( 1 + (-1.35 - 5.04i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.03 - 3.87i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.77 + 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.70 + 6.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.37 - 5.12i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (8.05 + 2.15i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (14.7 + 8.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.21 - 3.21i)T + 83iT^{2} \)
89 \( 1 + (-4.70 + 8.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.39 + 4.39i)T + 97iT^{2} \)
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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

−11.11820947200590656176303828058, −10.08285027224669334713296417703, −9.232809908783391707499447621982, −8.413561961771882216257406696706, −7.17495021814709398291331126656, −6.56077167034825585800854221831, −6.11421206591168022094162216691, −4.82021943477585327077840030153, −4.35697871000218477083399760524, −3.06161939274083660779881896517, 0.37360841386146663968048105529, 1.58036878624655993730211261894, 2.54075079466448551402164803539, 4.12257301360212260100566164559, 4.85247680857764489638284722716, 5.64927703513746662572046669634, 6.86419498256168488192661785162, 8.262433272251806886202669509441, 8.988607461462735224867502626004, 10.07930894906483397991872102372

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