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Properties

Label 2-735-105.104-c1-0-47
Degree $2$
Conductor $735$
Sign $0.737 - 0.674i$
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  + 2.50·2-s + (−1.47 + 0.900i)3-s + 4.28·4-s + (1.94 + 1.11i)5-s + (−3.71 + 2.25i)6-s + 5.74·8-s + (1.37 − 2.66i)9-s + (4.86 + 2.78i)10-s + 1.71i·11-s + (−6.34 + 3.86i)12-s + 0.360·13-s + (−3.87 + 0.104i)15-s + 5.82·16-s + 2.54i·17-s + (3.45 − 6.68i)18-s − 5.69i·19-s + ⋯
L(s)  = 1  + 1.77·2-s + (−0.854 + 0.519i)3-s + 2.14·4-s + (0.867 + 0.496i)5-s + (−1.51 + 0.922i)6-s + 2.03·8-s + (0.459 − 0.888i)9-s + (1.53 + 0.880i)10-s + 0.518i·11-s + (−1.83 + 1.11i)12-s + 0.100·13-s + (−0.999 + 0.0270i)15-s + 1.45·16-s + 0.616i·17-s + (0.814 − 1.57i)18-s − 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.737 - 0.674i$
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: \((2,\ 735,\ (\ :1/2),\ 0.737 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.49762 + 1.35820i\)
\(L(\frac12)\) \(\approx\) \(3.49762 + 1.35820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.47 - 0.900i)T \)
5 \( 1 + (-1.94 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 - 2.50T + 2T^{2} \)
11 \( 1 - 1.71iT - 11T^{2} \)
13 \( 1 - 0.360T + 13T^{2} \)
17 \( 1 - 2.54iT - 17T^{2} \)
19 \( 1 + 5.69iT - 19T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 - 3.76iT - 29T^{2} \)
31 \( 1 + 2.78iT - 31T^{2} \)
37 \( 1 - 3.30iT - 37T^{2} \)
41 \( 1 + 2.63T + 41T^{2} \)
43 \( 1 + 10.0iT - 43T^{2} \)
47 \( 1 + 5.82iT - 47T^{2} \)
53 \( 1 - 1.45T + 53T^{2} \)
59 \( 1 - 6.85T + 59T^{2} \)
61 \( 1 + 1.60iT - 61T^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 3.50iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 8.18T + 97T^{2} \)
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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

−10.67054813503709448512851476161, −10.11090202350520149933226298256, −8.982428815472715560137904771038, −7.15645045246975110599002214244, −6.61348137677561013302104576119, −5.77472804976404504543795818584, −5.11770583202441394053162843198, −4.23876998190453914418053364618, −3.21365339220731893632070196766, −1.96798010752846820027797206448, 1.48236915587359512214912940884, 2.66147148514196190282343840668, 4.09090132341790651969528331397, 5.01117232845337547161713572908, 5.81891243913069862385179669549, 6.17849622255523938294942781848, 7.20451151033197192236742150587, 8.283373632546319578916771666285, 9.765286314962459002882668201744, 10.62388676949624676535434990440

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