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Properties

Label 2-735-105.53-c1-0-64
Degree $2$
Conductor $735$
Sign $-0.987 + 0.156i$
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  + (0.340 − 1.26i)2-s + (−0.224 − 1.71i)3-s + (0.236 + 0.136i)4-s + (1.25 − 1.85i)5-s + (−2.25 − 0.299i)6-s + (2.11 − 2.11i)8-s + (−2.89 + 0.769i)9-s + (−1.92 − 2.21i)10-s + (−3.38 − 1.95i)11-s + (0.181 − 0.436i)12-s + (1.56 + 1.56i)13-s + (−3.46 − 1.73i)15-s + (−1.69 − 2.92i)16-s + (−2.58 + 0.693i)17-s + (−0.00887 + 3.94i)18-s + (1.61 − 0.930i)19-s + ⋯
L(s)  = 1  + (0.240 − 0.897i)2-s + (−0.129 − 0.991i)3-s + (0.118 + 0.0681i)4-s + (0.559 − 0.829i)5-s + (−0.921 − 0.122i)6-s + (0.746 − 0.746i)8-s + (−0.966 + 0.256i)9-s + (−0.609 − 0.701i)10-s + (−1.01 − 0.588i)11-s + (0.0523 − 0.125i)12-s + (0.434 + 0.434i)13-s + (−0.894 − 0.447i)15-s + (−0.422 − 0.731i)16-s + (−0.627 + 0.168i)17-s + (−0.00209 + 0.929i)18-s + (0.369 − 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145520 - 1.85263i\)
\(L(\frac12)\) \(\approx\) \(0.145520 - 1.85263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.224 + 1.71i)T \)
5 \( 1 + (-1.25 + 1.85i)T \)
7 \( 1 \)
good2 \( 1 + (-0.340 + 1.26i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (2.58 - 0.693i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.38 - 0.638i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.513T + 29T^{2} \)
31 \( 1 + (-4.29 + 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.498 + 1.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.259 + 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.55 - 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (2.79 - 0.749i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.81 + 6.81i)T - 97iT^{2} \)
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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

−10.19378981135474074572162643342, −9.120038351635735732465741344507, −8.259612909938894086712984401537, −7.41162340293300005230182926037, −6.36633110044301064350011183030, −5.50924023774055764089862800794, −4.37139838698977856038561933083, −2.92471724965621957122234043072, −2.05552966152401216736111255248, −0.896400643300317348223671670028, 2.26233902678390916604728167754, 3.36443089461967136663826112795, 4.85590959053636607958074123325, 5.39790052013163931334560047495, 6.33298874083334985503202201027, 7.09994752830258096025807431113, 8.081004833806927802117769302696, 9.082198346068010735167150854427, 10.21518163091331127050694364051, 10.57207165247447312300567268966

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