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Properties

Label 2-735-21.20-c1-0-17
Degree $2$
Conductor $735$
Sign $-0.846 - 0.532i$
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  + 2.06i·2-s + (1.71 − 0.263i)3-s − 2.25·4-s − 5-s + (0.543 + 3.53i)6-s − 0.527i·8-s + (2.86 − 0.902i)9-s − 2.06i·10-s + 4.69i·11-s + (−3.86 + 0.594i)12-s + 0.638i·13-s + (−1.71 + 0.263i)15-s − 3.42·16-s + 4.14·17-s + (1.86 + 5.90i)18-s − 0.897i·19-s + ⋯
L(s)  = 1  + 1.45i·2-s + (0.988 − 0.152i)3-s − 1.12·4-s − 0.447·5-s + (0.221 + 1.44i)6-s − 0.186i·8-s + (0.953 − 0.300i)9-s − 0.652i·10-s + 1.41i·11-s + (−1.11 + 0.171i)12-s + 0.177i·13-s + (−0.442 + 0.0680i)15-s − 0.855·16-s + 1.00·17-s + (0.438 + 1.39i)18-s − 0.205i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.846 - 0.532i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528044 + 1.83289i\)
\(L(\frac12)\) \(\approx\) \(0.528044 + 1.83289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.263i)T \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - 2.06iT - 2T^{2} \)
11 \( 1 - 4.69iT - 11T^{2} \)
13 \( 1 - 0.638iT - 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 + 0.897iT - 19T^{2} \)
23 \( 1 - 6.80iT - 23T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 - 2.33iT - 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + 4.10T + 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 - 6.80T + 47T^{2} \)
53 \( 1 + 2.26iT - 53T^{2} \)
59 \( 1 - 0.508T + 59T^{2} \)
61 \( 1 - 5.18iT - 61T^{2} \)
67 \( 1 - 4.82T + 67T^{2} \)
71 \( 1 - 1.22iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 - 9.08T + 79T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−10.36813891065589982575807854546, −9.495022987748007531922575895782, −8.787440073629549405036473963187, −7.83786526462499148075314392130, −7.36338078902726217389915884711, −6.75231734300900765829448913614, −5.44025768238860043435178444801, −4.52108998834680696802049282879, −3.43896968132818566866432366602, −1.89970134569021931799555606899, 0.934504296508082716994236450510, 2.38036067465314756345949220244, 3.34676002470749371424124562730, 3.85027119172388056218910590295, 5.11836765471511297157275868354, 6.60928948191357672361352873629, 7.80876079116213240613408984316, 8.603664077781970815595824734363, 9.219480948016281743416804603907, 10.39840174824512614043743404004

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