Properties

Label 2-735-105.2-c1-0-66
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: $\chi_{735} (422, \cdot )$
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

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

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