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Properties

Label 2-735-15.8-c1-0-20
Degree $2$
Conductor $735$
Sign $-0.525 - 0.850i$
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  + (0.929 + 0.929i)2-s + (−1.37 + 1.05i)3-s − 0.272i·4-s + (0.980 + 2.00i)5-s + (−2.25 − 0.299i)6-s + (2.11 − 2.11i)8-s + (0.782 − 2.89i)9-s + (−0.956 + 2.77i)10-s + 3.90i·11-s + (0.287 + 0.374i)12-s + (1.56 + 1.56i)13-s + (−3.46 − 1.73i)15-s + 3.38·16-s + (1.89 + 1.89i)17-s + (3.41 − 1.96i)18-s + 1.86i·19-s + ⋯
L(s)  = 1  + (0.657 + 0.657i)2-s + (−0.794 + 0.607i)3-s − 0.136i·4-s + (0.438 + 0.898i)5-s + (−0.921 − 0.122i)6-s + (0.746 − 0.746i)8-s + (0.260 − 0.965i)9-s + (−0.302 + 0.878i)10-s + 1.17i·11-s + (0.0828 + 0.108i)12-s + (0.434 + 0.434i)13-s + (−0.894 − 0.447i)15-s + 0.845·16-s + (0.459 + 0.459i)17-s + (0.805 − 0.462i)18-s + 0.426i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872146 + 1.56437i\)
\(L(\frac12)\) \(\approx\) \(0.872146 + 1.56437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 - 1.05i)T \)
5 \( 1 + (-0.980 - 2.00i)T \)
7 \( 1 \)
good2 \( 1 + (-0.929 - 0.929i)T + 2iT^{2} \)
11 \( 1 - 3.90iT - 11T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (-1.89 - 1.89i)T + 17iT^{2} \)
19 \( 1 - 1.86iT - 19T^{2} \)
23 \( 1 + (1.74 - 1.74i)T - 23iT^{2} \)
29 \( 1 + 0.513T + 29T^{2} \)
31 \( 1 + 8.58T + 31T^{2} \)
37 \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \)
41 \( 1 + 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (3.74 + 3.74i)T + 47iT^{2} \)
53 \( 1 + (1.36 - 1.36i)T - 53iT^{2} \)
59 \( 1 + 0.518T + 59T^{2} \)
61 \( 1 + 5.10T + 61T^{2} \)
67 \( 1 + (-6.40 + 6.40i)T - 67iT^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (-2.04 - 2.04i)T + 73iT^{2} \)
79 \( 1 - 5.05iT - 79T^{2} \)
83 \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
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

−10.67144331827193708003669386236, −9.858667111887748731755371308520, −9.393572433701031420302651069229, −7.61210611900770126351228471364, −6.89271858412305858760404774137, −6.06097683766206719098050342273, −5.52549735775640266556545844763, −4.41638775453337662936731618952, −3.63012754283557836034591649461, −1.73405412080803904461167239222, 0.881166449766289133112580944633, 2.19875000204521829206308063964, 3.50589588375780263017931916821, 4.72735550471742346433249456730, 5.47654291498943625511679102054, 6.21231565740050526714817556367, 7.58444409838158718897213480185, 8.267895436890432355777453450063, 9.233302464245293488375348577958, 10.54591210933691086395325644302

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