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Properties

Label 2-735-105.23-c1-0-71
Degree $2$
Conductor $735$
Sign $-0.805 - 0.592i$
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  + (1.69 − 0.455i)2-s + (−0.421 − 1.67i)3-s + (0.948 − 0.547i)4-s + (−2.12 + 0.710i)5-s + (−1.48 − 2.66i)6-s + (−1.12 + 1.12i)8-s + (−2.64 + 1.41i)9-s + (−3.28 + 2.17i)10-s + (−1.34 + 0.776i)11-s + (−1.32 − 1.36i)12-s + (−4.50 − 4.50i)13-s + (2.08 + 3.26i)15-s + (−2.49 + 4.32i)16-s + (−0.780 + 2.91i)17-s + (−3.84 + 3.61i)18-s + (−3.64 − 2.10i)19-s + ⋯
L(s)  = 1  + (1.20 − 0.322i)2-s + (−0.243 − 0.969i)3-s + (0.474 − 0.273i)4-s + (−0.948 + 0.317i)5-s + (−0.604 − 1.08i)6-s + (−0.397 + 0.397i)8-s + (−0.881 + 0.472i)9-s + (−1.03 + 0.687i)10-s + (−0.405 + 0.234i)11-s + (−0.381 − 0.393i)12-s + (−1.25 − 1.25i)13-s + (0.539 + 0.842i)15-s + (−0.623 + 1.08i)16-s + (−0.189 + 0.706i)17-s + (−0.907 + 0.851i)18-s + (−0.836 − 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0845952 + 0.257571i\)
\(L(\frac12)\) \(\approx\) \(0.0845952 + 0.257571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.421 + 1.67i)T \)
5 \( 1 + (2.12 - 0.710i)T \)
7 \( 1 \)
good2 \( 1 + (-1.69 + 0.455i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.34 - 0.776i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.50 + 4.50i)T + 13iT^{2} \)
17 \( 1 + (0.780 - 2.91i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.64 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.37 + 5.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + (-2.89 - 5.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.450 + 1.68i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (2.09 + 2.09i)T + 43iT^{2} \)
47 \( 1 + (-0.0489 + 0.0131i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.88 + 1.57i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.46 - 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.65 - 2.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.33 + 0.625i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.73iT - 71T^{2} \)
73 \( 1 + (-2.66 + 9.92i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.11 + 1.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.2 - 12.2i)T - 83iT^{2} \)
89 \( 1 + (-0.678 + 1.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−10.40516271426383127940922350495, −8.606316622247605627483161363357, −8.055685418393604527056418555174, −7.08550397704657953588528210305, −6.24931390574356370339269543145, −5.16380043541045108868841662211, −4.42670180806873972151097002032, −3.09113086636595281368226506116, −2.37510750294560294231991257048, −0.089967020896707498420264450732, 2.81182535691629057230596335152, 3.92495705115185635918959584448, 4.55413021549618742457331354263, 5.15838473039719876696357549217, 6.23857367876629784092842674010, 7.19705403860596172485373641521, 8.302814253817188102376970089161, 9.352238423462432551283112792069, 9.928964060212556893256059065677, 11.25261801651993924440294960418

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