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Properties

Label 2-735-49.8-c1-0-23
Degree $2$
Conductor $735$
Sign $0.930 + 0.367i$
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  + (1.02 + 1.28i)2-s + (−0.900 + 0.433i)3-s + (−0.155 + 0.679i)4-s + (−0.900 + 0.433i)5-s + (−1.47 − 0.712i)6-s + (−0.516 − 2.59i)7-s + (1.92 − 0.928i)8-s + (0.623 − 0.781i)9-s + (−1.47 − 0.712i)10-s + (−2.66 − 3.34i)11-s + (−0.155 − 0.679i)12-s + (−0.474 − 0.594i)13-s + (2.80 − 3.32i)14-s + (0.623 − 0.781i)15-s + (4.42 + 2.12i)16-s + (−0.265 − 1.16i)17-s + ⋯
L(s)  = 1  + (0.723 + 0.907i)2-s + (−0.520 + 0.250i)3-s + (−0.0775 + 0.339i)4-s + (−0.402 + 0.194i)5-s + (−0.604 − 0.290i)6-s + (−0.195 − 0.980i)7-s + (0.681 − 0.328i)8-s + (0.207 − 0.260i)9-s + (−0.467 − 0.225i)10-s + (−0.804 − 1.00i)11-s + (−0.0447 − 0.196i)12-s + (−0.131 − 0.164i)13-s + (0.748 − 0.887i)14-s + (0.160 − 0.201i)15-s + (1.10 + 0.532i)16-s + (−0.0645 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.930 + 0.367i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.930 + 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49151 - 0.283815i\)
\(L(\frac12)\) \(\approx\) \(1.49151 - 0.283815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.516 + 2.59i)T \)
good2 \( 1 + (-1.02 - 1.28i)T + (-0.445 + 1.94i)T^{2} \)
11 \( 1 + (2.66 + 3.34i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.474 + 0.594i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.265 + 1.16i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 - 1.64T + 19T^{2} \)
23 \( 1 + (-1.20 + 5.28i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.07 + 9.10i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 - 4.21T + 31T^{2} \)
37 \( 1 + (-1.27 - 5.58i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (6.12 - 2.94i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-4.61 - 2.22i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.12 - 5.17i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.137 - 0.604i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.28 + 1.58i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-0.132 - 0.579i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 + (-2.71 + 11.8i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.62 + 2.03i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 1.93T + 79T^{2} \)
83 \( 1 + (-7.50 + 9.41i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.60 - 7.02i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + 4.69T + 97T^{2} \)
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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

−10.49297557838995596955261652841, −9.683432841369340874669379269444, −8.158497707907155539516918796523, −7.57980679602527753647332001321, −6.58418925620560036690767947093, −5.98361294345708480629158437288, −4.90268530011738926493533171992, −4.22858305530169705941475732070, −3.07274792972762016141991539880, −0.67527739908887703062810409061, 1.71354571167351888299979048064, 2.74977503059724128707715059685, 3.89334553473021982861660667135, 5.05283292106721538602536246322, 5.49061928013307745055473070909, 6.99674384643241970313343665476, 7.70938291364980036428582281760, 8.795884632818370160407805361549, 9.857079537134266912086844642556, 10.71923057084321514191894741625

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