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Properties

Label 2-735-35.4-c1-0-6
Degree $2$
Conductor $735$
Sign $-0.860 - 0.508i$
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  + (−2.34 + 1.35i)2-s + (−0.866 − 0.5i)3-s + (2.67 − 4.62i)4-s + (1.55 + 1.60i)5-s + 2.70·6-s + 9.04i·8-s + (0.499 + 0.866i)9-s + (−5.82 − 1.67i)10-s + (−1 + 1.73i)11-s + (−4.62 + 2.67i)12-s + 0.921i·13-s + (−0.539 − 2.17i)15-s + (−6.91 − 11.9i)16-s + (0.933 + 0.539i)17-s + (−2.34 − 1.35i)18-s + (−1.53 − 2.66i)19-s + ⋯
L(s)  = 1  + (−1.65 + 0.957i)2-s + (−0.499 − 0.288i)3-s + (1.33 − 2.31i)4-s + (0.694 + 0.719i)5-s + 1.10·6-s + 3.19i·8-s + (0.166 + 0.288i)9-s + (−1.84 − 0.529i)10-s + (−0.301 + 0.522i)11-s + (−1.33 + 0.770i)12-s + 0.255i·13-s + (−0.139 − 0.560i)15-s + (−1.72 − 2.99i)16-s + (0.226 + 0.130i)17-s + (−0.553 − 0.319i)18-s + (−0.353 − 0.611i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122818 + 0.449393i\)
\(L(\frac12)\) \(\approx\) \(0.122818 + 0.449393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.55 - 1.60i)T \)
7 \( 1 \)
good2 \( 1 + (2.34 - 1.35i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.921iT - 13T^{2} \)
17 \( 1 + (-0.933 - 0.539i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.53 + 2.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.02 - 1.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + (3.87 - 6.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.38 + 5.41i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 - 6.52iT - 43T^{2} \)
47 \( 1 + (4.05 - 2.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.25 - 1.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.26 - 9.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.07 + 3.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.05 + 2.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (6.13 + 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.07 - 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 + (4.17 + 7.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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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

−10.52409607683815848280906095565, −9.781551637890227878563856538096, −9.084885568494164588007785140224, −8.034511010326409335527673061560, −7.24959356722669889070132957352, −6.57817029343841517408308982202, −5.93591167610160850980640458559, −4.92990605821477067178943768229, −2.53904077060630522528546973202, −1.40025572456451721798023501384, 0.46067974536858599037851040752, 1.72446712031515504860532106209, 2.95034239057800090968795802966, 4.27151213487889573171682976831, 5.69285306231101387497683584015, 6.66830132216378663337122286566, 7.978003960848846639238177318196, 8.463967549832640087383501128352, 9.417920528044752697954483223305, 10.03187649629197248318617903312

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