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Properties

Label 2-735-105.104-c1-0-19
Degree $2$
Conductor $735$
Sign $0.909 - 0.415i$
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  + 0.346·2-s + (−1.43 + 0.971i)3-s − 1.87·4-s + (−1.85 − 1.24i)5-s + (−0.496 + 0.336i)6-s − 1.34·8-s + (1.11 − 2.78i)9-s + (−0.644 − 0.430i)10-s + 0.537i·11-s + (2.69 − 1.82i)12-s − 3.81·13-s + (3.87 − 0.0267i)15-s + 3.29·16-s + 3.87i·17-s + (0.384 − 0.965i)18-s − 3.11i·19-s + ⋯
L(s)  = 1  + 0.244·2-s + (−0.827 + 0.561i)3-s − 0.939·4-s + (−0.831 − 0.555i)5-s + (−0.202 + 0.137i)6-s − 0.475·8-s + (0.370 − 0.928i)9-s + (−0.203 − 0.136i)10-s + 0.162i·11-s + (0.778 − 0.527i)12-s − 1.05·13-s + (0.999 − 0.00691i)15-s + 0.823·16-s + 0.940i·17-s + (0.0906 − 0.227i)18-s − 0.715i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670846 + 0.145929i\)
\(L(\frac12)\) \(\approx\) \(0.670846 + 0.145929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 - 0.971i)T \)
5 \( 1 + (1.85 + 1.24i)T \)
7 \( 1 \)
good2 \( 1 - 0.346T + 2T^{2} \)
11 \( 1 - 0.537iT - 11T^{2} \)
13 \( 1 + 3.81T + 13T^{2} \)
17 \( 1 - 3.87iT - 17T^{2} \)
19 \( 1 + 3.11iT - 19T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 + 8.41iT - 29T^{2} \)
31 \( 1 - 3.02iT - 31T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 - 8.56T + 41T^{2} \)
43 \( 1 - 4.12iT - 43T^{2} \)
47 \( 1 + 0.416iT - 47T^{2} \)
53 \( 1 - 4.28T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 0.282iT - 61T^{2} \)
67 \( 1 + 9.68iT - 67T^{2} \)
71 \( 1 + 1.01iT - 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 - 6.16T + 89T^{2} \)
97 \( 1 - 8.00T + 97T^{2} \)
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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

−10.40394864372787230492802371659, −9.565050730542741044240898755537, −8.895346779575757398096115770309, −7.945167152188496216227699484902, −6.86483183575028332561552050652, −5.67824782966662006314523109897, −4.72862988930375904743953036565, −4.41328434773090010694337357018, −3.22014379649876553952076161234, −0.74795180610306666105467322689, 0.64080166616684553540964195544, 2.72627179764388119742794627524, 3.99699251852414515289578962424, 4.97730356900506779561270379369, 5.67138342979663854263268451825, 7.05623380458951138456032252895, 7.41271163979674343965883504969, 8.535738640240843358370967694581, 9.503300163732710948576164527803, 10.51943302434780844499217498274

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