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Properties

Label 2-735-21.17-c1-0-48
Degree $2$
Conductor $735$
Sign $-0.683 - 0.729i$
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  + (0.686 − 0.396i)2-s + (−0.5 − 1.65i)3-s + (−0.686 + 1.18i)4-s + (−0.5 − 0.866i)5-s + (−1 − 0.939i)6-s + 2.67i·8-s + (−2.5 + 1.65i)9-s + (−0.686 − 0.396i)10-s + (−2.18 − 1.26i)11-s + (2.31 + 0.543i)12-s − 4.10i·13-s + (−1.18 + 1.26i)15-s + (−0.313 − 0.543i)16-s + (−2.18 + 3.78i)17-s + (−1.05 + 2.12i)18-s + (−3 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.485 − 0.280i)2-s + (−0.288 − 0.957i)3-s + (−0.343 + 0.594i)4-s + (−0.223 − 0.387i)5-s + (−0.408 − 0.383i)6-s + 0.944i·8-s + (−0.833 + 0.552i)9-s + (−0.216 − 0.125i)10-s + (−0.659 − 0.380i)11-s + (0.667 + 0.156i)12-s − 1.13i·13-s + (−0.306 + 0.325i)15-s + (−0.0784 − 0.135i)16-s + (−0.530 + 0.918i)17-s + (−0.249 + 0.501i)18-s + (−0.688 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.683 - 0.729i$
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: \((2,\ 735,\ (\ :1/2),\ -0.683 - 0.729i)\)

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 \( 1 + (0.5 + 1.65i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-0.686 + 0.396i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.18 + 1.26i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.37 - 4.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.939iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.37 - 5.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 + (-0.813 - 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.62 + 0.939i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.37 + 7.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 - 4.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.294iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (7.37 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−9.993520351858491363006438214363, −8.443609824283572369113302410814, −8.238031433223495609439805565836, −7.43605306084102291681294566740, −6.03882635594684108175454790410, −5.42473902684477284679254674194, −4.23488849194746780979303160692, −3.14665007656182472523434853285, −1.94704036782126069818120258769, 0, 2.42043711468817377514870868992, 4.00077382364220416236385425962, 4.49292449248453140819467458082, 5.43987791663344200418010095689, 6.36647405910844552689341622122, 7.14573780348509903700571580083, 8.593490888608196655393025055681, 9.351901992941828904131818680931, 10.12919248947150458968448577343

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