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Properties

Label 2-735-21.17-c1-0-40
Degree $2$
Conductor $735$
Sign $-0.997 - 0.0633i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 3·6-s − 1.73i·8-s + (1.5 + 2.59i)9-s + (1.5 + 0.866i)10-s + (−3 − 1.73i)11-s + (−1.5 + 0.866i)12-s + 1.73i·15-s + (2.49 + 4.33i)16-s + (3 − 5.19i)17-s + (−4.5 − 2.59i)18-s + (3 − 1.73i)19-s − 20-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (0.250 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 1.22·6-s − 0.612i·8-s + (0.5 + 0.866i)9-s + (0.474 + 0.273i)10-s + (−0.904 − 0.522i)11-s + (−0.433 + 0.250i)12-s + 0.447i·15-s + (0.624 + 1.08i)16-s + (0.727 − 1.26i)17-s + (−1.06 − 0.612i)18-s + (0.688 − 0.397i)19-s − 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.997 - 0.0633i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−9.954434108767726224492867325613, −8.925909160852142324607829471465, −8.067588576418741997280668476340, −7.43422218943838183854342870707, −6.72577597949370955525416273532, −5.57889991250662489813909274002, −4.86248568477507620396039223219, −3.19925851719562174403893913931, −1.24176101119156301268555931664, 0, 1.66028423997320428946064271532, 3.14761293328571175215733512142, 4.44823698710539668065695059809, 5.50162065303228152018525141007, 6.36079365394078231659565467589, 7.72174381128077248587751131451, 8.229660901280353154290682578055, 9.605916060702187968440168280532, 10.02602025289590951429925809664

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