Properties

Label 2-13e2-13.9-c3-0-24
Degree $2$
Conductor $169$
Sign $0.477 + 0.878i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
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  + (−2.5 + 4.33i)2-s + (3.5 − 6.06i)3-s + (−8.50 − 14.7i)4-s + 7·5-s + (17.5 + 30.3i)6-s + (−6.5 − 11.2i)7-s + 45.0·8-s + (−11 − 19.0i)9-s + (−17.5 + 30.3i)10-s + (−13 + 22.5i)11-s − 119.·12-s + 65·14-s + (24.5 − 42.4i)15-s + (−44.5 + 77.0i)16-s + (−38.5 − 66.6i)17-s + 109.·18-s + ⋯
L(s)  = 1  + (−0.883 + 1.53i)2-s + (0.673 − 1.16i)3-s + (−1.06 − 1.84i)4-s + 0.626·5-s + (1.19 + 2.06i)6-s + (−0.350 − 0.607i)7-s + 1.98·8-s + (−0.407 − 0.705i)9-s + (−0.553 + 0.958i)10-s + (−0.356 + 0.617i)11-s − 2.86·12-s + 1.24·14-s + (0.421 − 0.730i)15-s + (−0.695 + 1.20i)16-s + (−0.549 − 0.951i)17-s + 1.44·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.804048 - 0.478068i\)
\(L(\frac12)\) \(\approx\) \(0.804048 - 0.478068i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (2.5 - 4.33i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-3.5 + 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 7T + 125T^{2} \)
7 \( 1 + (6.5 + 11.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (13 - 22.5i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (38.5 + 66.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (63 + 109. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-48 + 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-41 + 71.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 196T + 2.97e4T^{2} \)
37 \( 1 + (65.5 - 113. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-168 + 290. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-100.5 - 174. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 105T + 1.03e5T^{2} \)
53 \( 1 + 432T + 1.48e5T^{2} \)
59 \( 1 + (147 + 254. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-28 - 48.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-239 + 413. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-4.5 - 7.79i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 98T + 3.89e5T^{2} \)
79 \( 1 - 1.30e3T + 4.93e5T^{2} \)
83 \( 1 - 308T + 5.71e5T^{2} \)
89 \( 1 + (595 - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-35 - 60.6i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−12.65952483754264354068649741060, −10.75639291940734256476841036414, −9.577893585456094749499755765687, −8.852583368604701797208425053984, −7.74579401140566999044386890038, −7.02486520333471652254689326860, −6.40532387260018605088522162402, −4.89114988219898330083302468811, −2.26740113364697859585444596896, −0.51709899821583641084620976354, 1.90016243719302224106282098731, 3.12907310242796333162048933211, 4.05678550095286923459644667792, 5.82733478085236609519846986499, 8.112211098299518116727171970876, 8.960116505072500222169624362845, 9.534210322362023116304212955912, 10.41953862586082272335670702926, 10.99918459226669365897682717624, 12.36032747335644965472207725732

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