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Properties

Label 2-735-7.4-c1-0-20
Degree $2$
Conductor $735$
Sign $-0.701 + 0.712i$
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  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s − 3·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−0.499 − 0.866i)12-s + 6·13-s + 0.999·15-s + (0.500 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.499 + 0.866i)18-s + (−4 − 6.92i)19-s + 20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.144 − 0.249i)12-s + 1.66·13-s + 0.258·15-s + (0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (−0.117 + 0.204i)18-s + (−0.917 − 1.58i)19-s + 0.223·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555565 - 1.32566i\)
\(L(\frac12)\) \(\approx\) \(0.555565 - 1.32566i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5 - 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−10.21513019721889744588714716247, −9.142671875229193952026188171827, −8.652693295096881803388888080875, −7.46699723572523875021634620936, −6.30232677462348876300017408378, −6.06215823738007843248542834061, −4.38811972194348669394918723538, −2.99376094750741457280557848510, −2.17265511341079055014304820807, −0.816880209091725251028262654467, 1.79144008333543833352681486237, 3.43584376871147869793226047999, 4.05053938266912190787151039746, 5.75599649755096734032464695844, 6.10687461110529749998803352213, 7.45495127667200245001312433880, 8.287735619010345288068790827385, 8.714492154223897861050509021309, 9.673084331152652731725376117625, 10.56324952502121165516736973958

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