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Properties

Label 2-735-105.2-c1-0-6
Degree $2$
Conductor $735$
Sign $-0.511 + 0.859i$
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  + (0.631 + 2.35i)2-s + (0.775 − 1.54i)3-s + (−3.42 + 1.97i)4-s + (−1.90 + 1.16i)5-s + (4.13 + 0.849i)6-s + (−3.36 − 3.36i)8-s + (−1.79 − 2.40i)9-s + (−3.94 − 3.76i)10-s + (−3.08 + 1.77i)11-s + (0.406 + 6.83i)12-s + (−1.28 + 1.28i)13-s + (0.323 + 3.85i)15-s + (1.85 − 3.21i)16-s + (−2.95 − 0.792i)17-s + (4.52 − 5.75i)18-s + (0.331 + 0.191i)19-s + ⋯
L(s)  = 1  + (0.446 + 1.66i)2-s + (0.447 − 0.894i)3-s + (−1.71 + 0.987i)4-s + (−0.853 + 0.520i)5-s + (1.68 + 0.346i)6-s + (−1.19 − 1.19i)8-s + (−0.599 − 0.800i)9-s + (−1.24 − 1.18i)10-s + (−0.928 + 0.536i)11-s + (0.117 + 1.97i)12-s + (−0.356 + 0.356i)13-s + (0.0834 + 0.996i)15-s + (0.463 − 0.803i)16-s + (−0.717 − 0.192i)17-s + (1.06 − 1.35i)18-s + (0.0761 + 0.0439i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.511 + 0.859i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264888 - 0.465878i\)
\(L(\frac12)\) \(\approx\) \(0.264888 - 0.465878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.775 + 1.54i)T \)
5 \( 1 + (1.90 - 1.16i)T \)
7 \( 1 \)
good2 \( 1 + (-0.631 - 2.35i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (3.08 - 1.77i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.28 - 1.28i)T - 13iT^{2} \)
17 \( 1 + (2.95 + 0.792i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.331 - 0.191i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.45 - 0.658i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.51T + 29T^{2} \)
31 \( 1 + (0.323 + 0.561i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.00 + 1.34i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 - 0.335i)T - 43iT^{2} \)
47 \( 1 + (0.751 + 2.80i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.815 - 3.04i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.81 - 6.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.45 + 9.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.31 - 12.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (3.17 + 0.849i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.21 - 1.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.973 + 0.973i)T + 83iT^{2} \)
89 \( 1 + (-1.51 + 2.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \)
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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

−11.14809832688002977233238345405, −9.737946881356011482712394015000, −8.705885030611450138443008726765, −7.904736778556952070657714635613, −7.44204121185233906245129043930, −6.79284993109030239164311531798, −5.94444825739268640915655118490, −4.78985918015585929381665945989, −3.80749555476518481037645644403, −2.46892177556016614309715935517, 0.21264553946293974657722889967, 2.19210651681173416036813716397, 3.22009145383271926307728663225, 3.98972926414686531237442533092, 4.80567632340523831631422057495, 5.56859209256448230303752320338, 7.59349343886984386815623928485, 8.480311085069125667019806050471, 9.204462069537745230389478599519, 10.06640688060969601548489403573

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