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Properties

Label 2-735-105.53-c1-0-7
Degree $2$
Conductor $735$
Sign $-0.855 + 0.517i$
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  + (−0.127 + 0.474i)2-s + (−1.01 + 1.40i)3-s + (1.52 + 0.879i)4-s + (−1.06 + 1.96i)5-s + (−0.536 − 0.659i)6-s + (−1.30 + 1.30i)8-s + (−0.942 − 2.84i)9-s + (−0.795 − 0.755i)10-s + (−2.31 − 1.33i)11-s + (−2.78 + 1.24i)12-s + (2.14 + 2.14i)13-s + (−1.67 − 3.49i)15-s + (1.30 + 2.26i)16-s + (−4.46 + 1.19i)17-s + (1.46 − 0.0850i)18-s + (−4.54 + 2.62i)19-s + ⋯
L(s)  = 1  + (−0.0898 + 0.335i)2-s + (−0.585 + 0.810i)3-s + (0.761 + 0.439i)4-s + (−0.477 + 0.878i)5-s + (−0.219 − 0.269i)6-s + (−0.461 + 0.461i)8-s + (−0.314 − 0.949i)9-s + (−0.251 − 0.238i)10-s + (−0.697 − 0.402i)11-s + (−0.802 + 0.359i)12-s + (0.596 + 0.596i)13-s + (−0.432 − 0.901i)15-s + (0.326 + 0.565i)16-s + (−1.08 + 0.290i)17-s + (0.346 − 0.0200i)18-s + (−1.04 + 0.601i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202034 - 0.723938i\)
\(L(\frac12)\) \(\approx\) \(0.202034 - 0.723938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.01 - 1.40i)T \)
5 \( 1 + (1.06 - 1.96i)T \)
7 \( 1 \)
good2 \( 1 + (0.127 - 0.474i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (2.31 + 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.14 - 2.14i)T + 13iT^{2} \)
17 \( 1 + (4.46 - 1.19i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.48 - 0.932i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 0.784i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \)
47 \( 1 + (2.80 - 10.4i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.62 - 6.05i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.0797 - 0.138i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.98 + 7.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (5.68 - 1.52i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.37 - 1.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 4.03i)T - 83iT^{2} \)
89 \( 1 + (1.97 + 3.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.86 - 1.86i)T - 97iT^{2} \)
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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

−11.00925804103770303129794051434, −10.37521252174180685892781773077, −9.083959143454857133046737771555, −8.270592340160178973124217330754, −7.29493853509914786313248206824, −6.38100673703002138017198480408, −5.89266014628848706840674108790, −4.37442200888671424235706257655, −3.53049878844429639454666623640, −2.40835387678155524564842906911, 0.40140256426715252003771313313, 1.69367447940742802822514459270, 2.84168083115877030071343159104, 4.57993528714951174610448202711, 5.40772973276511652118153147000, 6.42923902929801670945596074199, 7.14848277667663881184519195261, 8.121116707606458950401847557975, 8.907570418773422156590405862765, 10.15970006652018851655225485769

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