Properties

Label 2-735-105.23-c1-0-51
Degree $2$
Conductor $735$
Sign $-0.960 + 0.278i$
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  + (−1.46 + 0.391i)2-s + (−1.49 + 0.879i)3-s + (0.246 − 0.142i)4-s + (0.207 − 2.22i)5-s + (1.83 − 1.86i)6-s + (1.83 − 1.83i)8-s + (1.45 − 2.62i)9-s + (0.567 + 3.33i)10-s + (−0.791 + 0.457i)11-s + (−0.243 + 0.429i)12-s + (−3.07 − 3.07i)13-s + (1.64 + 3.50i)15-s + (−2.24 + 3.88i)16-s + (−0.311 + 1.16i)17-s + (−1.09 + 4.40i)18-s + (5.95 + 3.43i)19-s + ⋯
L(s)  = 1  + (−1.03 + 0.276i)2-s + (−0.861 + 0.507i)3-s + (0.123 − 0.0712i)4-s + (0.0929 − 0.995i)5-s + (0.749 − 0.762i)6-s + (0.648 − 0.648i)8-s + (0.484 − 0.874i)9-s + (0.179 + 1.05i)10-s + (−0.238 + 0.137i)11-s + (−0.0702 + 0.124i)12-s + (−0.854 − 0.854i)13-s + (0.425 + 0.905i)15-s + (−0.561 + 0.971i)16-s + (−0.0755 + 0.281i)17-s + (−0.258 + 1.03i)18-s + (1.36 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.960 + 0.278i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00602323 - 0.0423666i\)
\(L(\frac12)\) \(\approx\) \(0.00602323 - 0.0423666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.879i)T \)
5 \( 1 + (-0.207 + 2.22i)T \)
7 \( 1 \)
good2 \( 1 + (1.46 - 0.391i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.07 + 3.07i)T + 13iT^{2} \)
17 \( 1 + (0.311 - 1.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.95 - 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.505 + 1.88i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + (-2.31 - 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.207 - 0.774i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (10.1 - 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (10.6 + 2.85i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.533 - 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (-0.564 + 2.10i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.62 + 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.38 - 2.38i)T - 83iT^{2} \)
89 \( 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.843732788987862078697498014033, −9.363423425656006490373858547345, −8.274565638486428604904143000801, −7.66465647557721152334684710672, −6.54545739506384187406516854070, −5.35159347460071700268677380059, −4.78943836828609165560797322026, −3.57458005030178293157733097562, −1.34835627050699641026582499255, −0.03840922407830601684807147354, 1.62223095973399866824373691977, 2.78358441543944170295789521379, 4.59995926554156410770484578667, 5.50246506716623053236950904330, 6.66759884475115263214125180037, 7.37423022836014532863283256124, 8.014537293323903085351755113437, 9.497288766216840332449110332770, 9.778896513874044233489984860186, 10.80729836847074335689675266441

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