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Properties

Label 2-735-35.9-c1-0-22
Degree $2$
Conductor $735$
Sign $0.710 - 0.703i$
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  + (1.64 + 0.951i)2-s + (0.866 − 0.5i)3-s + (0.811 + 1.40i)4-s + (1.76 + 1.37i)5-s + 1.90·6-s − 0.719i·8-s + (0.499 − 0.866i)9-s + (1.59 + 3.94i)10-s + (−1 − 1.73i)11-s + (1.40 + 0.811i)12-s + 6.42i·13-s + (2.21 + 0.311i)15-s + (2.30 − 3.99i)16-s + (3.83 − 2.21i)17-s + (1.64 − 0.951i)18-s + (−1.21 + 2.10i)19-s + ⋯
L(s)  = 1  + (1.16 + 0.672i)2-s + (0.499 − 0.288i)3-s + (0.405 + 0.702i)4-s + (0.788 + 0.615i)5-s + 0.776·6-s − 0.254i·8-s + (0.166 − 0.288i)9-s + (0.504 + 1.24i)10-s + (−0.301 − 0.522i)11-s + (0.405 + 0.234i)12-s + 1.78i·13-s + (0.571 + 0.0803i)15-s + (0.576 − 0.998i)16-s + (0.930 − 0.537i)17-s + (0.388 − 0.224i)18-s + (−0.278 + 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.33928 + 1.37431i\)
\(L(\frac12)\) \(\approx\) \(3.33928 + 1.37431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-1.76 - 1.37i)T \)
7 \( 1 \)
good2 \( 1 + (-1.64 - 0.951i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.42iT - 13T^{2} \)
17 \( 1 + (-3.83 + 2.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.21 - 2.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.19 - 0.688i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + (2.59 + 4.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.59 + 3.80i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.23T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (2.38 + 1.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.95 - 4.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.05 - 12.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.42 - 5.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.38 + 1.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-1.36 + 0.785i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.42 - 4.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (-2.31 + 4.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−10.40445375437851824438719713430, −9.579334214199803192908426127045, −8.767738373608528988821012122537, −7.40414987304574378122065097970, −6.90007102887550922099442221965, −6.00980184592763287392320863387, −5.27674486505204063547703895597, −4.03149170371587856466023369849, −3.14158793766893021013526133102, −1.84022625232667932833102178031, 1.61983298445998794527616068455, 2.82659157255741781947265176840, 3.58975265742265949653764512912, 5.08959117491622891008271207345, 5.11863189919253504837708692393, 6.40225168885673565672473408638, 7.953997690493477084065231733723, 8.486221247061878869873975183837, 9.736474823772471452171853517403, 10.31570991099428787720837522148

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