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Properties

Label 2-735-105.53-c1-0-66
Degree $2$
Conductor $735$
Sign $-0.999 - 0.0199i$
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  + (0.565 − 2.10i)2-s + (0.871 − 1.49i)3-s + (−2.39 − 1.38i)4-s + (2.05 − 0.892i)5-s + (−2.66 − 2.68i)6-s + (−1.18 + 1.18i)8-s + (−1.48 − 2.60i)9-s + (−0.723 − 4.82i)10-s + (2.93 + 1.69i)11-s + (−4.16 + 2.38i)12-s + (−0.206 − 0.206i)13-s + (0.451 − 3.84i)15-s + (−0.935 − 1.62i)16-s + (−0.228 + 0.0612i)17-s + (−6.34 + 1.65i)18-s + (−4.60 + 2.65i)19-s + ⋯
L(s)  = 1  + (0.399 − 1.49i)2-s + (0.503 − 0.864i)3-s + (−1.19 − 0.692i)4-s + (0.916 − 0.399i)5-s + (−1.08 − 1.09i)6-s + (−0.419 + 0.419i)8-s + (−0.493 − 0.869i)9-s + (−0.228 − 1.52i)10-s + (0.883 + 0.510i)11-s + (−1.20 + 0.687i)12-s + (−0.0573 − 0.0573i)13-s + (0.116 − 0.993i)15-s + (−0.233 − 0.405i)16-s + (−0.0554 + 0.0148i)17-s + (−1.49 + 0.388i)18-s + (−1.05 + 0.609i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 - 0.0199i$
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: \((2,\ 735,\ (\ :1/2),\ -0.999 - 0.0199i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0250455 + 2.50690i\)
\(L(\frac12)\) \(\approx\) \(0.0250455 + 2.50690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.871 + 1.49i)T \)
5 \( 1 + (-2.05 + 0.892i)T \)
7 \( 1 \)
good2 \( 1 + (-0.565 + 2.10i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.206 + 0.206i)T + 13iT^{2} \)
17 \( 1 + (0.228 - 0.0612i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.60 - 2.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.93 - 1.85i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + (4.55 - 7.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.19 - 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (1.30 - 4.87i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.30 - 4.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.15 + 8.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 - 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 8.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (11.7 - 3.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.91 + 5.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.88 + 3.88i)T - 83iT^{2} \)
89 \( 1 + (1.00 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.26 + 2.26i)T - 97iT^{2} \)
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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

−9.996278871066123372016862450840, −9.224505247813364135247533299159, −8.672343859291546940223727502266, −7.25876888919121855225987327748, −6.39854823624611402384072292160, −5.22365738518743343163401946346, −4.08417478728709939323175787017, −2.99559432667353668121389358602, −1.94411639368294429206285391961, −1.22884397163908433755313092912, 2.32228809500640640220629891283, 3.70346453147263293749396291326, 4.67942342155078190013214326022, 5.56758336156714039338680382077, 6.36487337690109995951701339859, 7.12770352359415581340555043892, 8.246210839931173261726392729221, 9.035584877005930273059157203828, 9.544217425319453929347060097283, 10.78155306454438107574219562586

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