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Properties

Label 2-735-105.23-c1-0-58
Degree $2$
Conductor $735$
Sign $-0.0794 + 0.996i$
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 + (1.20 − 1.24i)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 + (−0.0946 − 2.99i)9-s + (−3.28 + 2.17i)10-s + (1.34 − 0.776i)11-s + (0.462 − 1.84i)12-s + (−4.50 − 4.50i)13-s + (1.67 − 3.49i)15-s + (−2.49 + 4.32i)16-s + (0.780 − 2.91i)17-s + (1.52 + 5.05i)18-s + (−3.64 − 2.10i)19-s + ⋯
L(s)  = 1  + (−1.20 + 0.322i)2-s + (0.695 − 0.718i)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 + (−0.0315 − 0.999i)9-s + (−1.03 + 0.687i)10-s + (0.405 − 0.234i)11-s + (0.133 − 0.531i)12-s + (−1.25 − 1.25i)13-s + (0.431 − 0.902i)15-s + (−0.623 + 1.08i)16-s + (0.189 − 0.706i)17-s + (0.359 + 1.19i)18-s + (−0.836 − 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0794 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0794 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0794 + 0.996i$
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: \((2,\ 735,\ (\ :1/2),\ -0.0794 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.686625 - 0.743521i\)
\(L(\frac12)\) \(\approx\) \(0.686625 - 0.743521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.20 + 1.24i)T \)
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

−9.818783878621285599098320898553, −9.171762710498050279890378917927, −8.574567059212459169685450004750, −7.62814322373806531183447496918, −7.05269750473440769795700479662, −6.03888015129100448631520279321, −4.85145177215012419707872911602, −3.17223885992701399192901610436, −1.96939388931497872443971055512, −0.70811165407948913238195635540, 1.85201025898581820778556160424, 2.47784721254893662609787735471, 4.13582505258620273236917735485, 5.03916655839917425729642491355, 6.41725668480173082843331215737, 7.42136926862994598363493082856, 8.418150103931612630733331795589, 9.133717743468601901387654937131, 9.700910152955908629899122101769, 10.27350749836349775507535821903

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