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Properties

Label 2-735-7.2-c1-0-22
Degree $2$
Conductor $735$
Sign $-0.605 + 0.795i$
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  + (1.11 − 1.93i)2-s + (0.5 + 0.866i)3-s + (−1.5 − 2.59i)4-s + (0.5 − 0.866i)5-s + 2.23·6-s − 2.23·8-s + (−0.499 + 0.866i)9-s + (−1.11 − 1.93i)10-s + (−3.23 − 5.60i)11-s + (1.50 − 2.59i)12-s + 4.47·13-s + 0.999·15-s + (0.499 − 0.866i)16-s + (1 + 1.73i)17-s + (1.11 + 1.93i)18-s + (1.23 − 2.14i)19-s + ⋯
L(s)  = 1  + (0.790 − 1.36i)2-s + (0.288 + 0.499i)3-s + (−0.750 − 1.29i)4-s + (0.223 − 0.387i)5-s + 0.912·6-s − 0.790·8-s + (−0.166 + 0.288i)9-s + (−0.353 − 0.612i)10-s + (−0.975 − 1.68i)11-s + (0.433 − 0.749i)12-s + 1.24·13-s + 0.258·15-s + (0.124 − 0.216i)16-s + (0.242 + 0.420i)17-s + (0.263 + 0.456i)18-s + (0.283 − 0.491i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11709 - 2.25345i\)
\(L(\frac12)\) \(\approx\) \(1.11709 - 2.25345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.93i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (3.23 + 5.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 2.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (0.763 + 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.47 + 6.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + (6.47 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.76 - 3.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.47 - 7.74i)T + (-29.5 + 51.0i)T^{2} \)
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 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 + (-6.23 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.47 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−10.37142958866323992239197483197, −9.466944200442640403829421425467, −8.620253693721461100630355227332, −7.77313276416401479404021236032, −5.86760155330411533562621027973, −5.47453635112413815676228785916, −4.19251955469449603336151155138, −3.45301951848110943180085046379, −2.56202131784614468843769230669, −1.06443774809138142836678398812, 1.98214080271177576832543989395, 3.42285039909483838064006738567, 4.56396405837541237210417247273, 5.48316120792426789059260460209, 6.38030925248051922713081065855, 7.09590094639384899377107285207, 7.79127432231226680472173586658, 8.497276518544809041650892267425, 9.758614276221420880039395597981, 10.55363048870955384470085255905

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