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Properties

Label 2-735-105.104-c1-0-25
Degree $2$
Conductor $735$
Sign $0.964 + 0.263i$
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  − 0.644·2-s + (−0.536 − 1.64i)3-s − 1.58·4-s + (2.15 − 0.607i)5-s + (0.346 + 1.06i)6-s + 2.31·8-s + (−2.42 + 1.76i)9-s + (−1.38 + 0.391i)10-s + 4.05i·11-s + (0.850 + 2.60i)12-s + 4.21·13-s + (−2.15 − 3.21i)15-s + 1.67·16-s + 2.17i·17-s + (1.56 − 1.14i)18-s + 4.47i·19-s + ⋯
L(s)  = 1  − 0.455·2-s + (−0.310 − 0.950i)3-s − 0.792·4-s + (0.962 − 0.271i)5-s + (0.141 + 0.433i)6-s + 0.817·8-s + (−0.807 + 0.589i)9-s + (−0.438 + 0.123i)10-s + 1.22i·11-s + (0.245 + 0.753i)12-s + 1.16·13-s + (−0.556 − 0.830i)15-s + 0.419·16-s + 0.527i·17-s + (0.368 − 0.268i)18-s + 1.02i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05598 - 0.141725i\)
\(L(\frac12)\) \(\approx\) \(1.05598 - 0.141725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.536 + 1.64i)T \)
5 \( 1 + (-2.15 + 0.607i)T \)
7 \( 1 \)
good2 \( 1 + 0.644T + 2T^{2} \)
11 \( 1 - 4.05iT - 11T^{2} \)
13 \( 1 - 4.21T + 13T^{2} \)
17 \( 1 - 2.17iT - 17T^{2} \)
19 \( 1 - 4.47iT - 19T^{2} \)
23 \( 1 - 0.644T + 23T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + 0.391iT - 31T^{2} \)
37 \( 1 + 4.26iT - 37T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 + 6.54iT - 43T^{2} \)
47 \( 1 - 7.80iT - 47T^{2} \)
53 \( 1 - 7.20T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 6.99iT - 61T^{2} \)
67 \( 1 + 8.73iT - 67T^{2} \)
71 \( 1 - 8.13iT - 71T^{2} \)
73 \( 1 - 5.23T + 73T^{2} \)
79 \( 1 - 3.75T + 79T^{2} \)
83 \( 1 - 5.27iT - 83T^{2} \)
89 \( 1 - 0.894T + 89T^{2} \)
97 \( 1 + 3.89T + 97T^{2} \)
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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

−10.26685485750024247676494478483, −9.424519815807165385163327755497, −8.611518887902392392874694983147, −7.901143269674738101159303895769, −6.87320173744114080417956399544, −5.90932877715756003726781460189, −5.16534434011343637024697352149, −3.93345352005970663302106130959, −2.07587326697422047877229980422, −1.16596855557703350431884872434, 0.871770798733542997494287384935, 2.92386472661303216857792552679, 3.97235272950174286243411118995, 5.11852835314046768164434646293, 5.78412853696473285859702316009, 6.74607348944889622228899588443, 8.312979736697597708593041905940, 8.898031752041004415034534663166, 9.508521498510532700446817292038, 10.35478255694063959368279886012

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