↑

Properties

Label 2-13e2-13.12-c7-0-1
Degree $2$
Conductor $169$
Sign $-0.691 + 0.722i$
Analytic cond. $52.7930$
Root an. cond. $7.26588$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6.93i·2-s − 64.5·3-s + 79.8·4-s + 33.5i·5-s − 447. i·6-s + 160. i·7-s + 1.44e3i·8-s + 1.97e3·9-s − 232.·10-s − 5.93e3i·11-s − 5.15e3·12-s − 1.11e3·14-s − 2.16e3i·15-s + 216.·16-s − 1.10e4·17-s + 1.37e4i·18-s + ⋯
L(s)  = 1  + 0.613i·2-s − 1.37·3-s + 0.623·4-s + 0.119i·5-s − 0.846i·6-s + 0.177i·7-s + 0.995i·8-s + 0.903·9-s − 0.0735·10-s − 1.34i·11-s − 0.860·12-s − 0.108·14-s − 0.165i·15-s + 0.0131·16-s − 0.545·17-s + 0.553i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(52.7930\)
Root analytic conductor: \(7.26588\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :7/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1951208612\)
\(L(\frac12)\) \(\approx\) \(0.1951208612\)
\(L(\frac{9}{2})\) 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 \( 1 - 6.93iT - 128T^{2} \)
3 \( 1 + 64.5T + 2.18e3T^{2} \)
5 \( 1 - 33.5iT - 7.81e4T^{2} \)
7 \( 1 - 160. iT - 8.23e5T^{2} \)
11 \( 1 + 5.93e3iT - 1.94e7T^{2} \)
17 \( 1 + 1.10e4T + 4.10e8T^{2} \)
19 \( 1 - 5.32e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.21e4T + 3.40e9T^{2} \)
29 \( 1 + 2.48e5T + 1.72e10T^{2} \)
31 \( 1 - 1.36e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.84e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.03e5iT - 1.94e11T^{2} \)
43 \( 1 + 5.32e5T + 2.71e11T^{2} \)
47 \( 1 - 2.87e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.37e6T + 1.17e12T^{2} \)
59 \( 1 + 5.14e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.27e6T + 3.14e12T^{2} \)
67 \( 1 + 4.00e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.78e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.26e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.10e6T + 1.92e13T^{2} \)
83 \( 1 + 4.96e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.03e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.33e7iT - 8.07e13T^{2} \)
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

−11.87939607238252503622461995002, −11.04267227696543718505712756525, −10.62628811389660892975840682098, −8.903574763373508045786336623847, −7.72065707097927129113549168547, −6.55255000499229356967152085109, −5.90743760282369114800761857865, −5.10757929805533291374538367010, −3.25205035311750106171456469274, −1.47071002758832592381223822614, 0.06306322134637245900103755383, 1.26881946581849785027849068562, 2.60369400992848539873172163138, 4.34463490518336322004174914691, 5.34443676119541928213572897449, 6.76360817079861157810504545463, 7.16509314235980259486323899098, 9.171773525521972674729425743566, 10.22308760910262943936908229892, 11.15866678959411019007119256335

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