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Properties

Label 2-735-15.8-c1-0-29
Degree $2$
Conductor $735$
Sign $-0.987 + 0.156i$
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  + (1.79 + 1.79i)2-s + (−0.491 + 1.66i)3-s + 4.47i·4-s + (1.87 + 1.21i)5-s + (−3.87 + 2.10i)6-s + (−4.45 + 4.45i)8-s + (−2.51 − 1.63i)9-s + (1.20 + 5.56i)10-s − 1.56i·11-s + (−7.43 − 2.19i)12-s + (−2.21 − 2.21i)13-s + (−2.93 + 2.52i)15-s − 7.09·16-s + (3.60 + 3.60i)17-s + (−1.59 − 7.46i)18-s − 1.68i·19-s + ⋯
L(s)  = 1  + (1.27 + 1.27i)2-s + (−0.283 + 0.958i)3-s + 2.23i·4-s + (0.840 + 0.541i)5-s + (−1.58 + 0.859i)6-s + (−1.57 + 1.57i)8-s + (−0.839 − 0.543i)9-s + (0.380 + 1.75i)10-s − 0.472i·11-s + (−2.14 − 0.634i)12-s + (−0.615 − 0.615i)13-s + (−0.757 + 0.652i)15-s − 1.77·16-s + (0.874 + 0.874i)17-s + (−0.375 − 1.75i)18-s − 0.385i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228548 - 2.89555i\)
\(L(\frac12)\) \(\approx\) \(0.228548 - 2.89555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.491 - 1.66i)T \)
5 \( 1 + (-1.87 - 1.21i)T \)
7 \( 1 \)
good2 \( 1 + (-1.79 - 1.79i)T + 2iT^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
13 \( 1 + (2.21 + 2.21i)T + 13iT^{2} \)
17 \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + (0.995 - 0.995i)T - 23iT^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + 2.74T + 31T^{2} \)
37 \( 1 + (-0.440 + 0.440i)T - 37iT^{2} \)
41 \( 1 + 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 + 5.47i)T + 43iT^{2} \)
47 \( 1 + (-3.69 - 3.69i)T + 47iT^{2} \)
53 \( 1 + (2.83 - 2.83i)T - 53iT^{2} \)
59 \( 1 - 5.54T + 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 + 3.61iT - 71T^{2} \)
73 \( 1 + (-5.89 - 5.89i)T + 73iT^{2} \)
79 \( 1 + 17.0iT - 79T^{2} \)
83 \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
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

−10.69675982673115750636723794326, −10.10588805302890442692389282075, −8.971805656536615450562556355734, −8.042942748705075368377386054157, −6.99297356785333684041743894836, −6.06593402351594308369596580385, −5.57407558722577523343255971016, −4.77508232139403703123397600366, −3.63482213660020235397679513282, −2.84384178645195781593787636722, 1.11646889777755200981954554334, 2.09116880101356473185023607315, 2.97578381838678500110809393773, 4.57761648057883555425456434813, 5.18400975202464361191721293244, 6.06995822275450717433782924829, 6.92606847057599738769554533195, 8.244902300653444230883254248418, 9.567044437834523521500293967371, 10.06275466929748663385991884460

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