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Properties

Label 2-735-15.8-c1-0-59
Degree $2$
Conductor $735$
Sign $0.621 - 0.783i$
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  + (1.54 + 1.54i)2-s + (1.73 + 0.00622i)3-s + 2.76i·4-s + (0.252 − 2.22i)5-s + (2.66 + 2.68i)6-s + (−1.18 + 1.18i)8-s + (2.99 + 0.0215i)9-s + (3.82 − 3.04i)10-s − 3.38i·11-s + (−0.0172 + 4.79i)12-s + (0.206 + 0.206i)13-s + (0.451 − 3.84i)15-s + 1.87·16-s + (−0.167 − 0.167i)17-s + (4.59 + 4.66i)18-s + 5.31i·19-s + ⋯
L(s)  = 1  + (1.09 + 1.09i)2-s + (0.999 + 0.00359i)3-s + 1.38i·4-s + (0.112 − 0.993i)5-s + (1.08 + 1.09i)6-s + (−0.419 + 0.419i)8-s + (0.999 + 0.00718i)9-s + (1.20 − 0.961i)10-s − 1.02i·11-s + (−0.00497 + 1.38i)12-s + (0.0573 + 0.0573i)13-s + (0.116 − 0.993i)15-s + 0.467·16-s + (−0.0406 − 0.0406i)17-s + (1.08 + 1.09i)18-s + 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.621 - 0.783i$
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: \((2,\ 735,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.42324 + 1.65460i\)
\(L(\frac12)\) \(\approx\) \(3.42324 + 1.65460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.00622i)T \)
5 \( 1 + (-0.252 + 2.22i)T \)
7 \( 1 \)
good2 \( 1 + (-1.54 - 1.54i)T + 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + (0.167 + 0.167i)T + 17iT^{2} \)
19 \( 1 - 5.31iT - 19T^{2} \)
23 \( 1 + (5.07 - 5.07i)T - 23iT^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + 9.11T + 31T^{2} \)
37 \( 1 + (5.27 - 5.27i)T - 37iT^{2} \)
41 \( 1 - 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (-3.56 - 3.56i)T + 47iT^{2} \)
53 \( 1 + (-3.55 + 3.55i)T - 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 6.80T + 61T^{2} \)
67 \( 1 + (-6.34 + 6.34i)T - 67iT^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (8.61 + 8.61i)T + 73iT^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (3.88 - 3.88i)T - 83iT^{2} \)
89 \( 1 + 2.00T + 89T^{2} \)
97 \( 1 + (2.26 - 2.26i)T - 97iT^{2} \)
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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

−10.27458805382077937932009432512, −9.382130525716317104261365767312, −8.415084346133816433691366781408, −7.951320097557036410666360180670, −7.00651918605507402109898069154, −5.82934118867656093057160128747, −5.27014199536872701894229802457, −3.98101330853441042407839077570, −3.53621182543553394277610101326, −1.68575441697198983924757633408, 1.98461611350171149219715928046, 2.53228198470093270572496203227, 3.63480227643932309745744927884, 4.30245066140575164170250424192, 5.47185855292539775617490221112, 6.82892873993024717091938653913, 7.48400712901328635533940779855, 8.717787081305165320533395350662, 9.788981243717325788164963181509, 10.35092553003104909792681860947

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