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Properties

Label 2-735-7.4-c1-0-5
Degree $2$
Conductor $735$
Sign $0.605 + 0.795i$
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  + (−1.20 − 2.09i)2-s + (−0.5 + 0.866i)3-s + (−1.91 + 3.31i)4-s + (−0.5 − 0.866i)5-s + 2.41·6-s + 4.41·8-s + (−0.499 − 0.866i)9-s + (−1.20 + 2.09i)10-s + (−1.41 + 2.44i)11-s + (−1.91 − 3.31i)12-s − 4.82·13-s + 0.999·15-s + (−1.49 − 2.59i)16-s + (3.82 − 6.63i)17-s + (−1.20 + 2.09i)18-s + (0.414 + 0.717i)19-s + ⋯
L(s)  = 1  + (−0.853 − 1.47i)2-s + (−0.288 + 0.499i)3-s + (−0.957 + 1.65i)4-s + (−0.223 − 0.387i)5-s + 0.985·6-s + 1.56·8-s + (−0.166 − 0.288i)9-s + (−0.381 + 0.661i)10-s + (−0.426 + 0.738i)11-s + (−0.552 − 0.957i)12-s − 1.33·13-s + 0.258·15-s + (−0.374 − 0.649i)16-s + (0.928 − 1.60i)17-s + (−0.284 + 0.492i)18-s + (0.0950 + 0.164i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585836 - 0.290415i\)
\(L(\frac12)\) \(\approx\) \(0.585836 - 0.290415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.41 - 2.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 + (-3.82 + 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.414 - 0.717i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.82 - 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (3.24 - 5.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (2.82 + 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.24 + 7.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.17 - 2.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + (-5.58 + 9.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.65T + 83T^{2} \)
89 \( 1 + (-2.65 - 4.60i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.48T + 97T^{2} \)
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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

−10.14188208027068298961424115730, −9.585506107029308139462348702159, −9.100705049676843155710732155307, −7.83890744590964923781720831229, −7.21410739840868236413808734438, −5.31247106158484299285649387837, −4.64949133199915553946549913601, −3.36131954724444709751343913716, −2.45220152870654412987479986272, −0.878063798946793224939747534544, 0.68101102281013593709002747963, 2.68234368547621746988347378805, 4.48498073078812778223524006802, 5.69145544353189354320740457565, 6.19133515535854095573213963610, 7.21212205618006407902763413819, 7.77393979109641428238731243817, 8.451068404806409614961644677026, 9.419476324572765473323289585076, 10.37697973638002951622160666226

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