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Properties

Label 2-735-15.8-c1-0-52
Degree $2$
Conductor $735$
Sign $0.966 + 0.256i$
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  + (1.59 + 1.59i)2-s + (−1.06 − 1.36i)3-s + 3.05i·4-s + (0.812 − 2.08i)5-s + (0.465 − 3.86i)6-s + (−1.68 + 1.68i)8-s + (−0.711 + 2.91i)9-s + (4.60 − 2.02i)10-s − 4.48i·11-s + (4.16 − 3.27i)12-s + (−1.08 − 1.08i)13-s + (−3.70 + 1.12i)15-s + 0.762·16-s + (1.49 + 1.49i)17-s + (−5.76 + 3.50i)18-s − 4.22i·19-s + ⋯
L(s)  = 1  + (1.12 + 1.12i)2-s + (−0.617 − 0.786i)3-s + 1.52i·4-s + (0.363 − 0.931i)5-s + (0.189 − 1.57i)6-s + (−0.595 + 0.595i)8-s + (−0.237 + 0.971i)9-s + (1.45 − 0.639i)10-s − 1.35i·11-s + (1.20 − 0.944i)12-s + (−0.300 − 0.300i)13-s + (−0.957 + 0.289i)15-s + 0.190·16-s + (0.363 + 0.363i)17-s + (−1.35 + 0.825i)18-s − 0.969i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.966 + 0.256i$
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: \((2,\ 735,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28974 - 0.299158i\)
\(L(\frac12)\) \(\approx\) \(2.28974 - 0.299158i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 + (-0.812 + 2.08i)T \)
7 \( 1 \)
good2 \( 1 + (-1.59 - 1.59i)T + 2iT^{2} \)
11 \( 1 + 4.48iT - 11T^{2} \)
13 \( 1 + (1.08 + 1.08i)T + 13iT^{2} \)
17 \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \)
19 \( 1 + 4.22iT - 19T^{2} \)
23 \( 1 + (-2.29 + 2.29i)T - 23iT^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + 1.06T + 31T^{2} \)
37 \( 1 + (-4.21 + 4.21i)T - 37iT^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \)
47 \( 1 + (-3.73 - 3.73i)T + 47iT^{2} \)
53 \( 1 + (6.11 - 6.11i)T - 53iT^{2} \)
59 \( 1 + 4.70T + 59T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + (0.416 - 0.416i)T - 67iT^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (-3.08 - 3.08i)T + 73iT^{2} \)
79 \( 1 - 6.67iT - 79T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + 3.51T + 89T^{2} \)
97 \( 1 + (5.60 - 5.60i)T - 97iT^{2} \)
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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

−10.58404613600674162220661862957, −9.163345431052390135059977123486, −8.209304688050724350629800713800, −7.58873858776272359191370694891, −6.45680387711525526387173129215, −5.91507442807908482468901500231, −5.18978212390787907444466260706, −4.40217409772943798569969253981, −2.87552516223889449311126848438, −0.959031572176501763045505791475, 1.81265787448418511698635042742, 2.99582172526188591572816547183, 3.90053005151240225858806358067, 4.79646227846543124752102943591, 5.58399005412419691277247361658, 6.54501903374442382856435319906, 7.58409498627404293378927720894, 9.389790151519984714206420587961, 9.963491992938604013813701324029, 10.55013149600934683508272542530

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