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Properties

Label 2-735-105.59-c1-0-30
Degree $2$
Conductor $735$
Sign $-0.937 + 0.347i$
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  + (−1.22 − 2.12i)2-s + (−1.5 − 0.866i)3-s + (−2.01 + 3.49i)4-s + (1.93 − 1.11i)5-s + 4.25i·6-s + 4.98·8-s + (1.5 + 2.59i)9-s + (−4.75 − 2.74i)10-s + (6.04 − 3.49i)12-s − 3.87·15-s + (−2.09 − 3.62i)16-s + (6.98 + 4.03i)17-s + (3.68 − 6.38i)18-s + (4.87 − 2.81i)19-s + 9.01i·20-s + ⋯
L(s)  = 1  + (−0.868 − 1.50i)2-s + (−0.866 − 0.499i)3-s + (−1.00 + 1.74i)4-s + (0.866 − 0.499i)5-s + 1.73i·6-s + 1.76·8-s + (0.5 + 0.866i)9-s + (−1.50 − 0.868i)10-s + (1.74 − 1.00i)12-s − 1.00·15-s + (−0.523 − 0.907i)16-s + (1.69 + 0.977i)17-s + (0.868 − 1.50i)18-s + (1.11 − 0.645i)19-s + 2.01i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.937 + 0.347i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.937 + 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142775 - 0.795559i\)
\(L(\frac12)\) \(\approx\) \(0.142775 - 0.795559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (1.22 + 2.12i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-6.98 - 4.03i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.87 + 2.81i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.79 + 8.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3.83 - 2.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-0.886 + 0.511i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.71 - 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.53 + 2.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.91 - 5.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.23158935625068985684678808576, −9.511853832316632351501482983295, −8.463935788508595501724887156430, −7.76648737899539410172773685128, −6.39992161475038004930020448395, −5.47471732655693110141979642371, −4.35325118123860787580682220881, −2.86851936606578786380473417713, −1.70828460455406947830107869831, −0.796010099272078644597129575200, 1.17430251416878947569117182063, 3.44607620159311935652214467208, 5.19161104283425694567017879680, 5.57925158341828652609310435789, 6.32019011940123350057820551871, 7.28698507290828210820162856557, 7.88587040173376196844083759924, 9.312867692920265844074725613309, 9.829021309719532256902164727724, 10.12504328951952188130842467795

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