Properties

Label 2-735-15.2-c1-0-54
Degree $2$
Conductor $735$
Sign $0.114 + 0.993i$
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  + (1.06 − 1.06i)2-s + (0.0150 − 1.73i)3-s − 0.285i·4-s + (2.03 + 0.933i)5-s + (−1.83 − 1.86i)6-s + (1.83 + 1.83i)8-s + (−2.99 − 0.0520i)9-s + (3.16 − 1.17i)10-s + 0.914i·11-s + (−0.493 − 0.00428i)12-s + (3.07 − 3.07i)13-s + (1.64 − 3.50i)15-s + 4.48·16-s + (0.850 − 0.850i)17-s + (−3.26 + 3.15i)18-s − 6.87i·19-s + ⋯
L(s)  = 1  + (0.755 − 0.755i)2-s + (0.00867 − 0.999i)3-s − 0.142i·4-s + (0.908 + 0.417i)5-s + (−0.749 − 0.762i)6-s + (0.648 + 0.648i)8-s + (−0.999 − 0.0173i)9-s + (1.00 − 0.371i)10-s + 0.275i·11-s + (−0.142 − 0.00123i)12-s + (0.854 − 0.854i)13-s + (0.425 − 0.905i)15-s + 1.12·16-s + (0.206 − 0.206i)17-s + (−0.768 + 0.742i)18-s − 1.57i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02692 - 1.80735i\)
\(L(\frac12)\) \(\approx\) \(2.02692 - 1.80735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0150 + 1.73i)T \)
5 \( 1 + (-2.03 - 0.933i)T \)
7 \( 1 \)
good2 \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \)
11 \( 1 - 0.914iT - 11T^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + (-0.850 + 0.850i)T - 17iT^{2} \)
19 \( 1 + 6.87iT - 19T^{2} \)
23 \( 1 + (1.38 + 1.38i)T + 23iT^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 - 4.63T + 31T^{2} \)
37 \( 1 + (-0.567 - 0.567i)T + 37iT^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (7.41 - 7.41i)T - 47iT^{2} \)
53 \( 1 + (-7.79 - 7.79i)T + 53iT^{2} \)
59 \( 1 + 9.88T + 59T^{2} \)
61 \( 1 + 1.06T + 61T^{2} \)
67 \( 1 + (5.00 + 5.00i)T + 67iT^{2} \)
71 \( 1 + 0.557iT - 71T^{2} \)
73 \( 1 + (1.54 - 1.54i)T - 73iT^{2} \)
79 \( 1 + 3.03iT - 79T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (1.58 + 1.58i)T + 97iT^{2} \)
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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

−10.58725421191516765970379322045, −9.400591880843830371346601002466, −8.376291810598105168038384554359, −7.51078305194462227579911908206, −6.53443358623149156783299395985, −5.69630302567408490498312277814, −4.71608270105165512260126853929, −3.16590347944261237540450944254, −2.57170408231199942456328789625, −1.36855993159039977686223093894, 1.66171997879247358416385934267, 3.53304991850672220243594222055, 4.32388154899016949095498528981, 5.34871018846601933356877121101, 5.90107334514144715051739436479, 6.62361565297320099283153361413, 8.093570576042217627613773906222, 8.883242952508167527661992782575, 9.923228141479592051857823661540, 10.24082135285503876350957183609

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