↑

Properties

Label 2-145-29.28-c3-0-16
Degree $2$
Conductor $145$
Sign $0.776 + 0.630i$
Analytic cond. $8.55527$
Root an. cond. $2.92494$
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  − 0.702i·2-s − 0.260i·3-s + 7.50·4-s − 5·5-s − 0.183·6-s + 0.749·7-s − 10.8i·8-s + 26.9·9-s + 3.51i·10-s − 20.4i·11-s − 1.95i·12-s + 28.8·13-s − 0.526i·14-s + 1.30i·15-s + 52.3·16-s + 85.6i·17-s + ⋯
L(s)  = 1  − 0.248i·2-s − 0.0501i·3-s + 0.938·4-s − 0.447·5-s − 0.0124·6-s + 0.0404·7-s − 0.481i·8-s + 0.997·9-s + 0.111i·10-s − 0.559i·11-s − 0.0470i·12-s + 0.616·13-s − 0.0100i·14-s + 0.0224i·15-s + 0.818·16-s + 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(8.55527\)
Root analytic conductor: \(2.92494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :3/2),\ 0.776 + 0.630i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96459 - 0.696703i\)
\(L(\frac12)\) \(\approx\) \(1.96459 - 0.696703i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
29 \( 1 + (-121. - 98.3i)T \)
good2 \( 1 + 0.702iT - 8T^{2} \)
3 \( 1 + 0.260iT - 27T^{2} \)
7 \( 1 - 0.749T + 343T^{2} \)
11 \( 1 + 20.4iT - 1.33e3T^{2} \)
13 \( 1 - 28.8T + 2.19e3T^{2} \)
17 \( 1 - 85.6iT - 4.91e3T^{2} \)
19 \( 1 + 157. iT - 6.85e3T^{2} \)
23 \( 1 - 67.9T + 1.21e4T^{2} \)
31 \( 1 + 184. iT - 2.97e4T^{2} \)
37 \( 1 + 286. iT - 5.06e4T^{2} \)
41 \( 1 - 108. iT - 6.89e4T^{2} \)
43 \( 1 - 352. iT - 7.95e4T^{2} \)
47 \( 1 - 508. iT - 1.03e5T^{2} \)
53 \( 1 + 667.T + 1.48e5T^{2} \)
59 \( 1 + 367.T + 2.05e5T^{2} \)
61 \( 1 - 513. iT - 2.26e5T^{2} \)
67 \( 1 - 94.0T + 3.00e5T^{2} \)
71 \( 1 + 38.3T + 3.57e5T^{2} \)
73 \( 1 - 721. iT - 3.89e5T^{2} \)
79 \( 1 - 365. iT - 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 796. iT - 7.04e5T^{2} \)
97 \( 1 + 1.02e3iT - 9.12e5T^{2} \)
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

−12.58747352050427769189063910003, −11.16029875725004025015457939037, −10.92703131803976247950906875980, −9.515644568050411326782948640782, −8.179903621324380910660509692492, −7.09249553575651914247692808832, −6.16278801827752945226931279604, −4.37514838713064248504465627965, −2.97863893211145720660837776092, −1.24047315129066032862278218114, 1.59464009757649104997359901346, 3.39540183106976530669982523088, 4.92524474904053088526760762831, 6.44578017515368798900884867782, 7.31923336758179348656853738149, 8.261813350099803113131794475750, 9.833955165949601445375476566734, 10.67513594276568221495133456618, 11.85146954055235277633961387748, 12.45054725576516745453364156653

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