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Properties

Label 2-735-15.8-c1-0-13
Degree $2$
Conductor $735$
Sign $0.848 - 0.529i$
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  + (−0.218 − 0.218i)2-s + (−0.354 + 1.69i)3-s − 1.90i·4-s + (−2.16 − 0.561i)5-s + (0.447 − 0.292i)6-s + (−0.852 + 0.852i)8-s + (−2.74 − 1.20i)9-s + (0.350 + 0.595i)10-s + 0.762i·11-s + (3.22 + 0.675i)12-s + (2.27 + 2.27i)13-s + (1.71 − 3.47i)15-s − 3.43·16-s + (3.43 + 3.43i)17-s + (0.337 + 0.862i)18-s + 1.63i·19-s + ⋯
L(s)  = 1  + (−0.154 − 0.154i)2-s + (−0.204 + 0.978i)3-s − 0.952i·4-s + (−0.967 − 0.251i)5-s + (0.182 − 0.119i)6-s + (−0.301 + 0.301i)8-s + (−0.916 − 0.400i)9-s + (0.110 + 0.188i)10-s + 0.229i·11-s + (0.932 + 0.194i)12-s + (0.629 + 0.629i)13-s + (0.443 − 0.896i)15-s − 0.859·16-s + (0.833 + 0.833i)17-s + (0.0796 + 0.203i)18-s + 0.375i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.848 - 0.529i$
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: \((2,\ 735,\ (\ :1/2),\ 0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973005 + 0.278435i\)
\(L(\frac12)\) \(\approx\) \(0.973005 + 0.278435i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.354 - 1.69i)T \)
5 \( 1 + (2.16 + 0.561i)T \)
7 \( 1 \)
good2 \( 1 + (0.218 + 0.218i)T + 2iT^{2} \)
11 \( 1 - 0.762iT - 11T^{2} \)
13 \( 1 + (-2.27 - 2.27i)T + 13iT^{2} \)
17 \( 1 + (-3.43 - 3.43i)T + 17iT^{2} \)
19 \( 1 - 1.63iT - 19T^{2} \)
23 \( 1 + (-5.40 + 5.40i)T - 23iT^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + (2.50 - 2.50i)T - 37iT^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \)
47 \( 1 + (-3.03 - 3.03i)T + 47iT^{2} \)
53 \( 1 + (-4.91 + 4.91i)T - 53iT^{2} \)
59 \( 1 + 7.69T + 59T^{2} \)
61 \( 1 + 4.39T + 61T^{2} \)
67 \( 1 + (-0.0345 + 0.0345i)T - 67iT^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (0.981 + 0.981i)T + 73iT^{2} \)
79 \( 1 - 4.23iT - 79T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 - 0.907T + 89T^{2} \)
97 \( 1 + (3.73 - 3.73i)T - 97iT^{2} \)
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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

−10.47385531415907536604203091874, −9.767202685703525220145397991061, −8.783467420054116596284553000747, −8.262370188094442209939654390314, −6.76608218808786235717167451302, −5.92674881436759499219022578462, −4.83074332167045934808838711698, −4.21859109863940756242305090190, −2.98433505944933382869310729932, −1.05589969009807048023660780048, 0.74976663171226113599993614991, 2.82368059340796884751378794352, 3.46006602446707188882180188580, 4.89358669354055669806521845672, 6.11541457239415154221810311899, 7.21889099566889094729524639645, 7.50897461701402685877149404236, 8.396258127667537167183262829885, 9.036759677779745245916964934902, 10.57526898169127787731668334127

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