↑

Properties

Label 2-735-105.23-c1-0-16
Degree $2$
Conductor $735$
Sign $-0.958 - 0.286i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.907 + 0.243i)2-s + (0.315 + 1.70i)3-s + (−0.967 + 0.558i)4-s + (2.12 + 0.695i)5-s + (−0.700 − 1.46i)6-s + (2.07 − 2.07i)8-s + (−2.80 + 1.07i)9-s + (−2.09 − 0.114i)10-s + (−0.630 + 0.363i)11-s + (−1.25 − 1.47i)12-s + (1.44 + 1.44i)13-s + (−0.515 + 3.83i)15-s + (−0.257 + 0.446i)16-s + (−1.90 + 7.09i)17-s + (2.28 − 1.65i)18-s + (−0.664 − 0.383i)19-s + ⋯
L(s)  = 1  + (−0.641 + 0.171i)2-s + (0.182 + 0.983i)3-s + (−0.483 + 0.279i)4-s + (0.950 + 0.311i)5-s + (−0.285 − 0.599i)6-s + (0.732 − 0.732i)8-s + (−0.933 + 0.357i)9-s + (−0.663 − 0.0363i)10-s + (−0.189 + 0.109i)11-s + (−0.362 − 0.425i)12-s + (0.400 + 0.400i)13-s + (−0.133 + 0.991i)15-s + (−0.0644 + 0.111i)16-s + (−0.460 + 1.71i)17-s + (0.537 − 0.390i)18-s + (−0.152 − 0.0879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.958 - 0.286i$
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: \((2,\ 735,\ (\ :1/2),\ -0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133513 + 0.913781i\)
\(L(\frac12)\) \(\approx\) \(0.133513 + 0.913781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.315 - 1.70i)T \)
5 \( 1 + (-2.12 - 0.695i)T \)
7 \( 1 \)
good2 \( 1 + (0.907 - 0.243i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.630 - 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 + (1.90 - 7.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.664 + 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.840 - 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (-0.209 - 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.63 + 6.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 + 5.15i)T + 43iT^{2} \)
47 \( 1 + (-6.79 + 1.82i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.26 + 1.41i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (6.90 + 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (-4.08 + 15.2i)T + (-63.2 - 36.5i)T^{2} \)
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 \( 1 + (5.62 - 5.62i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51627585208337747199538625469, −9.827920111192304807624012643763, −9.048881134899582369151245652644, −8.580537547763242899518278082897, −7.51368736420997454542226028882, −6.32657928071875402298772860932, −5.40466293089555929976367377456, −4.27939778690486404064569489601, −3.43312763033794374006143085063, −1.88118316147200263883471596259, 0.58726596011082533415582928749, 1.77969218798325623489184427822, 2.87943620376145473393672784487, 4.75553085887574272198252084643, 5.57720717461560390236602745545, 6.51801992517736883895185232677, 7.55229842948446932190102641305, 8.477400211264067961360277359902, 9.083805623826157653335927389960, 9.779942942250793960865730584507

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