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Properties

Label 2-735-105.23-c1-0-69
Degree $2$
Conductor $735$
Sign $-0.281 + 0.959i$
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  + (1.93 − 0.517i)2-s + (0.599 − 1.62i)3-s + (1.73 − 0.999i)4-s + (1.30 − 1.81i)5-s + (0.317 − 3.44i)6-s + (−2.28 − 1.94i)9-s + (1.58 − 4.18i)10-s + (−0.949 + 0.548i)11-s + (−0.585 − 3.41i)12-s + (−2.43 − 2.43i)13-s + (−2.16 − 3.21i)15-s + (−1.99 + 3.46i)16-s + (0.978 − 3.65i)17-s + (−5.41 − 2.58i)18-s + (6.67 + 3.85i)19-s + (0.449 − 4.44i)20-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)2-s + (0.346 − 0.938i)3-s + (0.866 − 0.499i)4-s + (0.584 − 0.811i)5-s + (0.129 − 1.40i)6-s + (−0.760 − 0.649i)9-s + (0.501 − 1.32i)10-s + (−0.286 + 0.165i)11-s + (−0.169 − 0.985i)12-s + (−0.676 − 0.676i)13-s + (−0.558 − 0.829i)15-s + (−0.499 + 0.866i)16-s + (0.237 − 0.886i)17-s + (−1.27 − 0.609i)18-s + (1.53 + 0.884i)19-s + (0.100 − 0.994i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.281 + 0.959i$
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: \((2,\ 735,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09086 - 2.79119i\)
\(L(\frac12)\) \(\approx\) \(2.09086 - 2.79119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.599 + 1.62i)T \)
5 \( 1 + (-1.30 + 1.81i)T \)
7 \( 1 \)
good2 \( 1 + (-1.93 + 0.517i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.949 - 0.548i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.43 + 2.43i)T + 13iT^{2} \)
17 \( 1 + (-0.978 + 3.65i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.67 - 3.85i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.15 - 4.29i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.66T + 29T^{2} \)
31 \( 1 + (-1.73 - 3i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 4.71i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.89iT - 41T^{2} \)
43 \( 1 + (-1.89 - 1.89i)T + 43iT^{2} \)
47 \( 1 + (3.65 - 0.978i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.80 + 0.750i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.24 - 7.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.11 - 0.567i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + (2.58 - 9.65i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.3 - 5.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.34 - 5.34i)T - 83iT^{2} \)
89 \( 1 + (-8.44 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 - 2.43i)T - 97iT^{2} \)
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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

−10.12481032021981169511333749444, −9.319453707964826123750144739446, −8.265169150506873243327142657270, −7.45775556683436935279338495239, −6.30186358036590259498092513871, −5.38697225866169219643076925229, −4.89766350382330375562913854731, −3.34759962138589454806974041856, −2.58121660507848731643836444942, −1.24936558235640218963221052177, 2.57096309320958123949871810393, 3.20452751152991810447014531309, 4.37472055351778689421940855552, 5.08355068172706067744609928818, 6.00194043241585709972128930401, 6.80593335945391098317002032888, 7.85120730130154041815782686192, 9.166485624839321319965628651800, 9.803742303467096887971839076556, 10.65302056052512280430025979651

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