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Properties

Label 2-735-21.17-c1-0-15
Degree $2$
Conductor $735$
Sign $0.618 - 0.785i$
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  + (0.334 − 0.192i)2-s + (0.139 − 1.72i)3-s + (−0.925 + 1.60i)4-s + (0.5 + 0.866i)5-s + (−0.286 − 0.603i)6-s + 1.48i·8-s + (−2.96 − 0.480i)9-s + (0.334 + 0.192i)10-s + (2.20 + 1.27i)11-s + (2.63 + 1.82i)12-s + 3.06i·13-s + (1.56 − 0.742i)15-s + (−1.56 − 2.71i)16-s + (−3.23 + 5.59i)17-s + (−1.08 + 0.410i)18-s + (1.03 − 0.597i)19-s + ⋯
L(s)  = 1  + (0.236 − 0.136i)2-s + (0.0803 − 0.996i)3-s + (−0.462 + 0.801i)4-s + (0.223 + 0.387i)5-s + (−0.116 − 0.246i)6-s + 0.525i·8-s + (−0.987 − 0.160i)9-s + (0.105 + 0.0609i)10-s + (0.663 + 0.383i)11-s + (0.761 + 0.525i)12-s + 0.850i·13-s + (0.404 − 0.191i)15-s + (−0.391 − 0.677i)16-s + (−0.783 + 1.35i)17-s + (−0.254 + 0.0968i)18-s + (0.237 − 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27521 + 0.618694i\)
\(L(\frac12)\) \(\approx\) \(1.27521 + 0.618694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.139 + 1.72i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.334 + 0.192i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.20 - 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.03 + 0.597i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.64 + 1.52i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.77iT - 29T^{2} \)
31 \( 1 + (-5.95 - 3.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.77 - 3.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 + (1.61 + 2.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.4 + 6.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.98 - 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.08 + 4.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.75 + 3.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.921iT - 71T^{2} \)
73 \( 1 + (0.256 + 0.148i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 + 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + (-9.41 - 16.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−10.75243596725436027060671523842, −9.362676133899695776007188815549, −8.720519131587940018304862441114, −7.940885291469233401283921981199, −6.85636208195146379096769013420, −6.45154709528852046637671755308, −5.00798182717321861838533896091, −3.91509505314613276235723853697, −2.82962568341726579332874879934, −1.66077274604201853884182567256, 0.70445157837215350510106780357, 2.71030551154905711172024375601, 4.06025043844838018109508703996, 4.78840712404640844274310475456, 5.63330152934349891895444480993, 6.35654884363763699758979237465, 7.82000863803402632289785210149, 8.914588426365951983262006970719, 9.449912051903229674703498816870, 10.03734684809399490566695109467

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