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Properties

Label 2-735-105.23-c1-0-66
Degree $2$
Conductor $735$
Sign $0.799 + 0.601i$
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  + (2.35 − 0.631i)2-s + (1.54 + 0.775i)3-s + (3.42 − 1.97i)4-s + (0.0540 − 2.23i)5-s + (4.13 + 0.849i)6-s + (3.36 − 3.36i)8-s + (1.79 + 2.40i)9-s + (−1.28 − 5.30i)10-s + (−3.08 + 1.77i)11-s + (6.83 − 0.406i)12-s + (−1.28 − 1.28i)13-s + (1.81 − 3.42i)15-s + (1.85 − 3.21i)16-s + (−0.792 + 2.95i)17-s + (5.75 + 4.52i)18-s + (−0.331 − 0.191i)19-s + ⋯
L(s)  = 1  + (1.66 − 0.446i)2-s + (0.894 + 0.447i)3-s + (1.71 − 0.987i)4-s + (0.0241 − 0.999i)5-s + (1.68 + 0.346i)6-s + (1.19 − 1.19i)8-s + (0.599 + 0.800i)9-s + (−0.406 − 1.67i)10-s + (−0.928 + 0.536i)11-s + (1.97 − 0.117i)12-s + (−0.356 − 0.356i)13-s + (0.469 − 0.883i)15-s + (0.463 − 0.803i)16-s + (−0.192 + 0.717i)17-s + (1.35 + 1.06i)18-s + (−0.0761 − 0.0439i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.799 + 0.601i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.799 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.59133 - 1.53442i\)
\(L(\frac12)\) \(\approx\) \(4.59133 - 1.53442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 - 0.775i)T \)
5 \( 1 + (-0.0540 + 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (-2.35 + 0.631i)T + (1.73 - i)T^{2} \)
11 \( 1 + (3.08 - 1.77i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.28 + 1.28i)T + 13iT^{2} \)
17 \( 1 + (0.792 - 2.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.331 + 0.191i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.658 + 2.45i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + (0.323 + 0.561i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.34 + 5.00i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 + 0.335i)T + 43iT^{2} \)
47 \( 1 + (2.80 - 0.751i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.04 + 0.815i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (-12.3 - 3.31i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.06iT - 71T^{2} \)
73 \( 1 + (-0.849 + 3.17i)T + (-63.2 - 36.5i)T^{2} \)
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

−10.37433494819173380184427041291, −9.685519445337878563101614119218, −8.488868988382008205293892660805, −7.78730699133631540782771616469, −6.46605356952020749402191871009, −5.24419010228761397195053235748, −4.74209753470522569694622369276, −3.92770586734981815708925979591, −2.81082443367346819022158755380, −1.87528326042320519549139945855, 2.33402017192151560249593917201, 2.99706676039841897354222883836, 3.86113869674634745987500835180, 5.02916844415141714858973146840, 6.07933222697746027199166522828, 6.92083988107910636190317726337, 7.44650343611972430188968139455, 8.386333846741498156165546396165, 9.669166551044939829252483868463, 10.67027989553052414343540106097

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