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Properties

Label 2-735-105.2-c1-0-24
Degree $2$
Conductor $735$
Sign $-0.610 - 0.792i$
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  + (0.582 + 2.17i)2-s + (−1.71 − 0.245i)3-s + (−2.64 + 1.52i)4-s + (2.21 + 0.337i)5-s + (−0.465 − 3.86i)6-s + (−1.68 − 1.68i)8-s + (2.87 + 0.840i)9-s + (0.552 + 4.99i)10-s + (3.88 − 2.24i)11-s + (4.91 − 1.97i)12-s + (1.08 − 1.08i)13-s + (−3.70 − 1.12i)15-s + (−0.381 + 0.660i)16-s + (2.04 + 0.548i)17-s + (−0.150 + 6.74i)18-s + (3.66 + 2.11i)19-s + ⋯
L(s)  = 1  + (0.411 + 1.53i)2-s + (−0.989 − 0.141i)3-s + (−1.32 + 0.764i)4-s + (0.988 + 0.151i)5-s + (−0.189 − 1.57i)6-s + (−0.595 − 0.595i)8-s + (0.959 + 0.280i)9-s + (0.174 + 1.58i)10-s + (1.17 − 0.676i)11-s + (1.41 − 0.569i)12-s + (0.300 − 0.300i)13-s + (−0.957 − 0.289i)15-s + (−0.0952 + 0.165i)16-s + (0.496 + 0.133i)17-s + (−0.0354 + 1.58i)18-s + (0.839 + 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.610 - 0.792i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.610 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763593 + 1.55268i\)
\(L(\frac12)\) \(\approx\) \(0.763593 + 1.55268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.245i)T \)
5 \( 1 + (-2.21 - 0.337i)T \)
7 \( 1 \)
good2 \( 1 + (-0.582 - 2.17i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.08 + 1.08i)T - 13iT^{2} \)
17 \( 1 + (-2.04 - 0.548i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.66 - 2.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 - 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (0.530 + 0.918i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.75 - 1.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 + 2.00i)T - 43iT^{2} \)
47 \( 1 + (1.36 + 5.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.23 - 8.34i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.88 - 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.152 - 0.569i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (-4.22 - 1.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.78 - 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + (1.75 - 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.60 - 5.60i)T + 97iT^{2} \)
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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

−10.61056084275691798501650886853, −9.743761876269962763304874596408, −8.796244526303556318416573739392, −7.75645218175924711055134100898, −6.86654036713931319466733075834, −6.08730562508107705113458783515, −5.75947989566961062915529543227, −4.82377323722243773734392855809, −3.61397784199899335201151217687, −1.42824718405595737897844123946, 1.09915173467598163268815659478, 1.99195120146078770660950770133, 3.49888751209369558908228910941, 4.50434892452275969887716586493, 5.26912422401162992074141355441, 6.29462038698732607061710282297, 7.18334603521328019787479340848, 9.011965798280584726007065445789, 9.656602013049643880116381373121, 10.18595677071522300271380792092

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