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Properties

Label 2-735-105.89-c1-0-58
Degree $2$
Conductor $735$
Sign $-0.205 + 0.978i$
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  + (0.866 − 1.5i)2-s + (0.724 − 1.57i)3-s + (−0.5 − 0.866i)4-s + (0.358 + 2.20i)5-s + (−1.73 − 2.44i)6-s + 1.73·8-s + (−1.94 − 2.28i)9-s + (3.62 + 1.37i)10-s + (2.44 − 1.41i)11-s + (−1.72 + 0.158i)12-s + 4·13-s + (3.73 + 1.03i)15-s + (2.49 − 4.33i)16-s + (−2.44 + 1.41i)17-s + (−5.10 + 0.949i)18-s + ⋯
L(s)  = 1  + (0.612 − 1.06i)2-s + (0.418 − 0.908i)3-s + (−0.250 − 0.433i)4-s + (0.160 + 0.987i)5-s + (−0.707 − 0.999i)6-s + 0.612·8-s + (−0.649 − 0.760i)9-s + (1.14 + 0.434i)10-s + (0.738 − 0.426i)11-s + (−0.497 + 0.0458i)12-s + 1.10·13-s + (0.963 + 0.267i)15-s + (0.624 − 1.08i)16-s + (−0.594 + 0.342i)17-s + (−1.20 + 0.223i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71793 - 2.11666i\)
\(L(\frac12)\) \(\approx\) \(1.71793 - 2.11666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.724 + 1.57i)T \)
5 \( 1 + (-0.358 - 2.20i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-2.44 + 1.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (2.44 - 1.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (8.48 - 4.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 + (-2.44 - 1.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.48 + 4.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.24 - 2.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.56463840414735325221855699901, −9.349280930957885025951491865586, −8.446762471742849798705196225556, −7.43188377942543029943182344630, −6.59200307922928769753017485506, −5.82575900359298957649952101241, −4.09475637191309815281492426865, −3.35035653529814721748392111824, −2.43847119916631803215484171248, −1.37530460747743157402968343138, 1.71761675171600041691113280698, 3.67518180464445118210757641545, 4.37757962405988752325907230917, 5.25630454875332259232597679959, 5.93430631177754586338892968637, 7.07332549810623077061408416880, 8.038745875652973587577703574245, 8.974083533910311817357427827965, 9.381415738071790960731790451653, 10.61209635589324081199130768962

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