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Properties

Label 2-735-105.53-c1-0-59
Degree $2$
Conductor $735$
Sign $-0.431 + 0.902i$
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  + (0.293 − 1.09i)2-s + (1.70 − 0.275i)3-s + (0.622 + 0.359i)4-s + (−1.70 − 1.44i)5-s + (0.199 − 1.95i)6-s + (2.17 − 2.17i)8-s + (2.84 − 0.943i)9-s + (−2.08 + 1.44i)10-s + (−4.50 − 2.60i)11-s + (1.16 + 0.442i)12-s + (−3.24 − 3.24i)13-s + (−3.31 − 2.00i)15-s + (−1.02 − 1.77i)16-s + (−1.15 + 0.309i)17-s + (−0.197 − 3.39i)18-s + (1.14 − 0.660i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.773i)2-s + (0.987 − 0.159i)3-s + (0.311 + 0.179i)4-s + (−0.763 − 0.646i)5-s + (0.0813 − 0.796i)6-s + (0.769 − 0.769i)8-s + (0.949 − 0.314i)9-s + (−0.657 + 0.456i)10-s + (−1.35 − 0.784i)11-s + (0.335 + 0.127i)12-s + (−0.900 − 0.900i)13-s + (−0.856 − 0.516i)15-s + (−0.255 − 0.443i)16-s + (−0.279 + 0.0749i)17-s + (−0.0465 − 0.799i)18-s + (0.262 − 0.151i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.431 + 0.902i$
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: \((2,\ 735,\ (\ :1/2),\ -0.431 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19791 - 1.90068i\)
\(L(\frac12)\) \(\approx\) \(1.19791 - 1.90068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.275i)T \)
5 \( 1 + (1.70 + 1.44i)T \)
7 \( 1 \)
good2 \( 1 + (-0.293 + 1.09i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (4.50 + 2.60i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.24 + 3.24i)T + 13iT^{2} \)
17 \( 1 + (1.15 - 0.309i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.14 + 0.660i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.68 - 2.06i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 + (-0.852 + 1.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.33 - 0.626i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (0.281 + 0.281i)T + 43iT^{2} \)
47 \( 1 + (1.24 - 4.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.28 - 4.79i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.908 - 1.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.23 - 2.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.89 - 10.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 + (-1.82 + 0.489i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.96 - 5.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.46 - 5.46i)T - 83iT^{2} \)
89 \( 1 + (-4.71 - 8.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.06 - 3.06i)T - 97iT^{2} \)
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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

−10.24627173540489782693076337467, −9.265173727478842540792352376064, −8.224943229024192308291358456063, −7.74278159542753296710639013411, −6.97506958284951667638430834865, −5.27511332209724782289692182229, −4.33294969165373834979155442591, −3.10066942828736314322759082262, −2.70175803622173489917750465424, −0.988976750801982431405318638694, 2.17350267522414652021623211989, 2.96480894666525395871027582971, 4.47898338746721044971291244021, 5.05674225144974866440237062429, 6.71939392495532332601577449165, 7.18528889816704099005205504393, 7.81492254779390245997670284476, 8.667001983116403159137974062937, 9.878315792215140101473496457350, 10.50860700850093884842578796782

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