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Properties

Label 2-735-105.23-c1-0-4
Degree $2$
Conductor $735$
Sign $-0.999 + 0.0275i$
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  + (1.09 − 0.293i)2-s + (−0.275 + 1.70i)3-s + (−0.622 + 0.359i)4-s + (0.398 − 2.20i)5-s + (0.199 + 1.95i)6-s + (−2.17 + 2.17i)8-s + (−2.84 − 0.943i)9-s + (−0.208 − 2.52i)10-s + (−4.50 + 2.60i)11-s + (−0.442 − 1.16i)12-s + (−3.24 − 3.24i)13-s + (3.65 + 1.28i)15-s + (−1.02 + 1.77i)16-s + (−0.309 + 1.15i)17-s + (−3.39 − 0.197i)18-s + (−1.14 − 0.660i)19-s + ⋯
L(s)  = 1  + (0.773 − 0.207i)2-s + (−0.159 + 0.987i)3-s + (−0.311 + 0.179i)4-s + (0.178 − 0.983i)5-s + (0.0813 + 0.796i)6-s + (−0.769 + 0.769i)8-s + (−0.949 − 0.314i)9-s + (−0.0660 − 0.797i)10-s + (−1.35 + 0.784i)11-s + (−0.127 − 0.335i)12-s + (−0.900 − 0.900i)13-s + (0.942 + 0.332i)15-s + (−0.255 + 0.443i)16-s + (−0.0749 + 0.279i)17-s + (−0.799 − 0.0465i)18-s + (−0.262 − 0.151i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 + 0.0275i$
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: \((2,\ 735,\ (\ :1/2),\ -0.999 + 0.0275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00470809 - 0.341123i\)
\(L(\frac12)\) \(\approx\) \(0.00470809 - 0.341123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.275 - 1.70i)T \)
5 \( 1 + (-0.398 + 2.20i)T \)
7 \( 1 \)
good2 \( 1 + (-1.09 + 0.293i)T + (1.73 - i)T^{2} \)
11 \( 1 + (4.50 - 2.60i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.24 + 3.24i)T + 13iT^{2} \)
17 \( 1 + (0.309 - 1.15i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.14 + 0.660i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.06 - 7.68i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + (-0.852 - 1.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.626 + 2.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.82iT - 41T^{2} \)
43 \( 1 + (0.281 + 0.281i)T + 43iT^{2} \)
47 \( 1 + (4.63 - 1.24i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.79 - 1.28i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.908 - 1.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.23 + 2.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.8 + 2.89i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (0.489 - 1.82i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.96 - 5.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.46 + 5.46i)T - 83iT^{2} \)
89 \( 1 + (4.71 - 8.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.06 - 3.06i)T - 97iT^{2} \)
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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

−10.79042880519005839775767960132, −9.882533256179536868822950673987, −9.293325851725163226750782062064, −8.346108147269577452244756440438, −7.55618258164496064391116341202, −5.66691665109059186317971729444, −5.23638502752164161249765286565, −4.61969351164014320501792733371, −3.58092613893572043984862773571, −2.45357795851892426964967497944, 0.12829724631585852863019963895, 2.30575400518590611479959688388, 3.16076518711726905420607245691, 4.65294619410240065825698305123, 5.57463554994920199263048559004, 6.37956633338585830960634278804, 7.04586214393160820071123179997, 7.999141095533780947220469034899, 9.015328823054234606113361921775, 10.11664874004982019007627848627

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