Properties

Label 2-735-5.4-c1-0-8
Degree $2$
Conductor $735$
Sign $-0.990 + 0.139i$
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  + 1.90i·2-s + i·3-s − 1.62·4-s + (−0.311 − 2.21i)5-s − 1.90·6-s + 0.719i·8-s − 9-s + (4.21 − 0.592i)10-s + 2·11-s − 1.62i·12-s + 6.42i·13-s + (2.21 − 0.311i)15-s − 4.61·16-s + 4.42i·17-s − 1.90i·18-s − 2.42·19-s + ⋯
L(s)  = 1  + 1.34i·2-s + 0.577i·3-s − 0.811·4-s + (−0.139 − 0.990i)5-s − 0.776·6-s + 0.254i·8-s − 0.333·9-s + (1.33 − 0.187i)10-s + 0.603·11-s − 0.468i·12-s + 1.78i·13-s + (0.571 − 0.0803i)15-s − 1.15·16-s + 1.07i·17-s − 0.448i·18-s − 0.557·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0857336 - 1.22641i\)
\(L(\frac12)\) \(\approx\) \(0.0857336 - 1.22641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (0.311 + 2.21i)T \)
7 \( 1 \)
good2 \( 1 - 1.90iT - 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 6.42iT - 13T^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 - 1.37iT - 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 + 7.61iT - 37T^{2} \)
41 \( 1 - 8.23T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 - 2.75iT - 47T^{2} \)
53 \( 1 - 9.18iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 + 2.75iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 1.57iT - 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
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

−10.87992880697985188169226920061, −9.342642690807313162078080782909, −9.122101043121774676322138266014, −8.255555820969375763094854391087, −7.34665925427669369105334865590, −6.32654004481742673525509851657, −5.66718469964334565201056478374, −4.49077159180770507744289603237, −4.04515006699795042465510280448, −1.86718664373272650919099690877, 0.62562193882476938977424344886, 2.19643238624277705314844881968, 3.02123399643056820234641464575, 3.85275289790133415522046864631, 5.37098360459983443326350085548, 6.55701797311922521624238793483, 7.26971181215233741514312860580, 8.268746775796355345442153788664, 9.399060042606303329317380467346, 10.25980881559575675385165251927

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