↑

Properties

Label 2-45-15.8-c3-0-4
Degree $2$
Conductor $45$
Sign $-0.0739 + 0.997i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.287 − 0.287i)2-s − 7.83i·4-s + (−9.93 − 5.12i)5-s + (15.0 − 15.0i)7-s + (−4.55 + 4.55i)8-s + (1.38 + 4.33i)10-s − 27.9i·11-s + (2.49 + 2.49i)13-s − 8.66·14-s − 60.0·16-s + (67.9 + 67.9i)17-s + 95.2i·19-s + (−40.1 + 77.8i)20-s + (−8.06 + 8.06i)22-s + (121. − 121. i)23-s + ⋯
L(s)  = 1  + (−0.101 − 0.101i)2-s − 0.979i·4-s + (−0.888 − 0.457i)5-s + (0.812 − 0.812i)7-s + (−0.201 + 0.201i)8-s + (0.0438 + 0.137i)10-s − 0.767i·11-s + (0.0533 + 0.0533i)13-s − 0.165·14-s − 0.938·16-s + (0.968 + 0.968i)17-s + 1.14i·19-s + (−0.448 + 0.870i)20-s + (−0.0781 + 0.0781i)22-s + (1.10 − 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0739 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0739 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.0739 + 0.997i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.0739 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.757039 - 0.815228i\)
\(L(\frac12)\) \(\approx\) \(0.757039 - 0.815228i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (9.93 + 5.12i)T \)
good2 \( 1 + (0.287 + 0.287i)T + 8iT^{2} \)
7 \( 1 + (-15.0 + 15.0i)T - 343iT^{2} \)
11 \( 1 + 27.9iT - 1.33e3T^{2} \)
13 \( 1 + (-2.49 - 2.49i)T + 2.19e3iT^{2} \)
17 \( 1 + (-67.9 - 67.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 95.2iT - 6.85e3T^{2} \)
23 \( 1 + (-121. + 121. i)T - 1.21e4iT^{2} \)
29 \( 1 + 99.0T + 2.43e4T^{2} \)
31 \( 1 + 28.7T + 2.97e4T^{2} \)
37 \( 1 + (-271. + 271. i)T - 5.06e4iT^{2} \)
41 \( 1 + 453. iT - 6.89e4T^{2} \)
43 \( 1 + (-30.5 - 30.5i)T + 7.95e4iT^{2} \)
47 \( 1 + (-254. - 254. i)T + 1.03e5iT^{2} \)
53 \( 1 + (224. - 224. i)T - 1.48e5iT^{2} \)
59 \( 1 - 483.T + 2.05e5T^{2} \)
61 \( 1 + 264.T + 2.26e5T^{2} \)
67 \( 1 + (498. - 498. i)T - 3.00e5iT^{2} \)
71 \( 1 - 609. iT - 3.57e5T^{2} \)
73 \( 1 + (74.6 + 74.6i)T + 3.89e5iT^{2} \)
79 \( 1 + 406. iT - 4.93e5T^{2} \)
83 \( 1 + (652. - 652. i)T - 5.71e5iT^{2} \)
89 \( 1 + 139.T + 7.04e5T^{2} \)
97 \( 1 + (-557. + 557. i)T - 9.12e5iT^{2} \)
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

−14.84391280826047376207728079558, −14.19887490998199363047178176575, −12.63463989819496362976453607806, −11.21123389258118878941017814021, −10.48768336185751299273195673218, −8.793830161244021545943872017857, −7.58274411916516140025321407946, −5.68508930045068881475223014020, −4.12627927451318625251653159480, −1.02963253810936528977934635325, 2.98706897347848056854270181873, 4.80686238397433738414283075142, 7.12508071744936662643248217318, 7.998179048780012605282840568336, 9.316176515344609851365752154682, 11.36420247434226690203447279984, 11.87368664525218213213159676160, 13.17459860767315819739796683986, 14.82814341807490311470855640525, 15.51997154690549536141991179800

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