↑

Properties

Label 2-735-15.2-c1-0-38
Degree $2$
Conductor $735$
Sign $0.980 - 0.196i$
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  + (1.13 − 1.13i)2-s + (0.811 + 1.53i)3-s − 0.574i·4-s + (1.66 − 1.48i)5-s + (2.65 + 0.815i)6-s + (1.61 + 1.61i)8-s + (−1.68 + 2.48i)9-s + (0.206 − 3.58i)10-s + 5.63i·11-s + (0.878 − 0.465i)12-s + (−2.65 + 2.65i)13-s + (3.63 + 1.34i)15-s + 4.81·16-s + (2.15 − 2.15i)17-s + (0.908 + 4.72i)18-s − 5.72i·19-s + ⋯
L(s)  = 1  + (0.802 − 0.802i)2-s + (0.468 + 0.883i)3-s − 0.287i·4-s + (0.746 − 0.665i)5-s + (1.08 + 0.332i)6-s + (0.571 + 0.571i)8-s + (−0.560 + 0.827i)9-s + (0.0652 − 1.13i)10-s + 1.69i·11-s + (0.253 − 0.134i)12-s + (−0.736 + 0.736i)13-s + (0.937 + 0.347i)15-s + 1.20·16-s + (0.523 − 0.523i)17-s + (0.214 + 1.11i)18-s − 1.31i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.98987 + 0.297333i\)
\(L(\frac12)\) \(\approx\) \(2.98987 + 0.297333i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.811 - 1.53i)T \)
5 \( 1 + (-1.66 + 1.48i)T \)
7 \( 1 \)
good2 \( 1 + (-1.13 + 1.13i)T - 2iT^{2} \)
11 \( 1 - 5.63iT - 11T^{2} \)
13 \( 1 + (2.65 - 2.65i)T - 13iT^{2} \)
17 \( 1 + (-2.15 + 2.15i)T - 17iT^{2} \)
19 \( 1 + 5.72iT - 19T^{2} \)
23 \( 1 + (2.88 + 2.88i)T + 23iT^{2} \)
29 \( 1 - 2.13T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + (-2.57 - 2.57i)T + 37iT^{2} \)
41 \( 1 + 9.28iT - 41T^{2} \)
43 \( 1 + (-2.42 + 2.42i)T - 43iT^{2} \)
47 \( 1 + (0.633 - 0.633i)T - 47iT^{2} \)
53 \( 1 + (5.02 + 5.02i)T + 53iT^{2} \)
59 \( 1 + 8.09T + 59T^{2} \)
61 \( 1 - 9.62T + 61T^{2} \)
67 \( 1 + (-4.11 - 4.11i)T + 67iT^{2} \)
71 \( 1 + 3.23iT - 71T^{2} \)
73 \( 1 + (-7.66 + 7.66i)T - 73iT^{2} \)
79 \( 1 - 7.81iT - 79T^{2} \)
83 \( 1 + (-1.26 - 1.26i)T + 83iT^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + (6.53 + 6.53i)T + 97iT^{2} \)
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

−10.27589094575904950407079032136, −9.730246550871003750019074370364, −9.043586637263745620627095173257, −7.961362675715092328334250226875, −6.90003835732416434107314089670, −5.27577544969525996139217107238, −4.73582248776853236576275953575, −4.13643159993801746870677597677, −2.64356767826180771179430015164, −2.02981809172300686555366912615, 1.32120928956175365239971825425, 2.87483542878827080110891800701, 3.71844620885446558223842464776, 5.47677046403083672671141907942, 5.96818693298804798910561970290, 6.55813227413127063711617561252, 7.73254419942740239227261537046, 8.129749128315692022385514309155, 9.531993311977898227969389945209, 10.27736108373811140661172546796

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