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Properties

Label 2-735-105.2-c1-0-38
Degree $2$
Conductor $735$
Sign $0.467 - 0.883i$
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  + (0.243 + 0.907i)2-s + (1.31 + 1.12i)3-s + (0.967 − 0.558i)4-s + (−1.66 − 1.49i)5-s + (−0.700 + 1.46i)6-s + (2.07 + 2.07i)8-s + (0.470 + 2.96i)9-s + (0.949 − 1.87i)10-s + (0.630 − 0.363i)11-s + (1.90 + 0.352i)12-s + (1.44 − 1.44i)13-s + (−0.515 − 3.83i)15-s + (−0.257 + 0.446i)16-s + (7.09 + 1.90i)17-s + (−2.57 + 1.14i)18-s + (0.664 + 0.383i)19-s + ⋯
L(s)  = 1  + (0.171 + 0.641i)2-s + (0.760 + 0.649i)3-s + (0.483 − 0.279i)4-s + (−0.744 − 0.667i)5-s + (−0.285 + 0.599i)6-s + (0.732 + 0.732i)8-s + (0.156 + 0.987i)9-s + (0.300 − 0.592i)10-s + (0.189 − 0.109i)11-s + (0.549 + 0.101i)12-s + (0.400 − 0.400i)13-s + (−0.133 − 0.991i)15-s + (−0.0644 + 0.111i)16-s + (1.71 + 0.460i)17-s + (−0.606 + 0.270i)18-s + (0.152 + 0.0879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.467 - 0.883i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.467 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04748 + 1.23270i\)
\(L(\frac12)\) \(\approx\) \(2.04748 + 1.23270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.31 - 1.12i)T \)
5 \( 1 + (1.66 + 1.49i)T \)
7 \( 1 \)
good2 \( 1 + (-0.243 - 0.907i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.630 + 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 + (-7.09 - 1.90i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.664 - 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 - 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (-0.209 - 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.08 + 1.63i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (1.82 + 6.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.41 + 5.26i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.807 - 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.78 - 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.84 + 6.90i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.06iT - 71T^{2} \)
73 \( 1 + (15.2 + 4.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.80 + 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 - 1.83i)T + 83iT^{2} \)
89 \( 1 + (6.94 - 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.62 + 5.62i)T + 97iT^{2} \)
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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

−10.40459985901351928521464575071, −9.663569007839633966793893869421, −8.552901118481944738198883944338, −7.921566911804014927819006817075, −7.35514271544837288803416903544, −5.89818332173063550915894983235, −5.21354964429344675715301866280, −4.10214051216931545225433797697, −3.19399521587424465921776703603, −1.55814736767984137400939940628, 1.33690265011778417429817349914, 2.65823901930652635304659670004, 3.39664039388346749544393021316, 4.19850708039696620486856502917, 6.05786360132021545774848130364, 6.99349118129230173919382698725, 7.62733160466879568831806164942, 8.245792305629536495880855469635, 9.538701778860736118145838393766, 10.27873761271148690961174670138

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