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Properties

Label 2-735-15.8-c1-0-5
Degree $2$
Conductor $735$
Sign $-0.788 - 0.615i$
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  + (−1.54 − 1.54i)2-s + (0.00622 + 1.73i)3-s + 2.76i·4-s + (−0.252 + 2.22i)5-s + (2.66 − 2.68i)6-s + (1.18 − 1.18i)8-s + (−2.99 + 0.0215i)9-s + (3.82 − 3.04i)10-s + 3.38i·11-s + (−4.79 + 0.0172i)12-s + (0.206 + 0.206i)13-s + (−3.84 − 0.423i)15-s + 1.87·16-s + (0.167 + 0.167i)17-s + (4.66 + 4.59i)18-s + 5.31i·19-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + (0.00359 + 0.999i)3-s + 1.38i·4-s + (−0.112 + 0.993i)5-s + (1.08 − 1.09i)6-s + (0.419 − 0.419i)8-s + (−0.999 + 0.00718i)9-s + (1.20 − 0.961i)10-s + 1.02i·11-s + (−1.38 + 0.00497i)12-s + (0.0573 + 0.0573i)13-s + (−0.994 − 0.109i)15-s + 0.467·16-s + (0.0406 + 0.0406i)17-s + (1.09 + 1.08i)18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.788 - 0.615i$
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: \((2,\ 735,\ (\ :1/2),\ -0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133407 + 0.387493i\)
\(L(\frac12)\) \(\approx\) \(0.133407 + 0.387493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.00622 - 1.73i)T \)
5 \( 1 + (0.252 - 2.22i)T \)
7 \( 1 \)
good2 \( 1 + (1.54 + 1.54i)T + 2iT^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + (-0.167 - 0.167i)T + 17iT^{2} \)
19 \( 1 - 5.31iT - 19T^{2} \)
23 \( 1 + (-5.07 + 5.07i)T - 23iT^{2} \)
29 \( 1 - 2.84T + 29T^{2} \)
31 \( 1 + 9.11T + 31T^{2} \)
37 \( 1 + (5.27 - 5.27i)T - 37iT^{2} \)
41 \( 1 + 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (3.56 + 3.56i)T + 47iT^{2} \)
53 \( 1 + (3.55 - 3.55i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 6.80T + 61T^{2} \)
67 \( 1 + (-6.34 + 6.34i)T - 67iT^{2} \)
71 \( 1 - 3.95iT - 71T^{2} \)
73 \( 1 + (8.61 + 8.61i)T + 73iT^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (-3.88 + 3.88i)T - 83iT^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
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

−10.45931245138630593604448798176, −10.13228176694646853200856334888, −9.312751215665819534610381906200, −8.503070587676093275815289241477, −7.58015403220516834573495950976, −6.45734774955433849745464764042, −5.13882875279640768367759967148, −3.84169173037418610366962842041, −3.03824574454351297733498295707, −1.94286404098067410665082925267, 0.31368522718244031645412604643, 1.42255326612726444049073895070, 3.28183996486012843620439350417, 5.15418429375398943671289844704, 5.81320008831842390871879708699, 6.82849437620938944708754250464, 7.50874741049154428709245624439, 8.320877885776042905508491978717, 8.923080674882377504399020876979, 9.438171219270806659478772361792

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