↑

Properties

Label 2-13e2-13.12-c3-0-20
Degree $2$
Conductor $169$
Sign $0.832 + 0.554i$
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  + 2.56i·2-s − 3.68·3-s + 1.43·4-s + 0.561i·5-s − 9.43i·6-s − 18.1i·7-s + 24.1i·8-s − 13.4·9-s − 1.43·10-s − 64.7i·11-s − 5.30·12-s + 46.5·14-s − 2.06i·15-s − 50.4·16-s + 25.5·17-s − 34.3i·18-s + ⋯
L(s)  = 1  + 0.905i·2-s − 0.709·3-s + 0.179·4-s + 0.0502i·5-s − 0.642i·6-s − 0.981i·7-s + 1.06i·8-s − 0.497·9-s − 0.0454·10-s − 1.77i·11-s − 0.127·12-s + 0.888·14-s − 0.0356i·15-s − 0.787·16-s + 0.364·17-s − 0.450i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07475 - 0.325410i\)
\(L(\frac12)\) \(\approx\) \(1.07475 - 0.325410i\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.56iT - 8T^{2} \)
3 \( 1 + 3.68T + 27T^{2} \)
5 \( 1 - 0.561iT - 125T^{2} \)
7 \( 1 + 18.1iT - 343T^{2} \)
11 \( 1 + 64.7iT - 1.33e3T^{2} \)
17 \( 1 - 25.5T + 4.91e3T^{2} \)
19 \( 1 + 107. iT - 6.85e3T^{2} \)
23 \( 1 + 73.2T + 1.21e4T^{2} \)
29 \( 1 - 175.T + 2.43e4T^{2} \)
31 \( 1 + 113. iT - 2.97e4T^{2} \)
37 \( 1 + 114. iT - 5.06e4T^{2} \)
41 \( 1 + 69.6iT - 6.89e4T^{2} \)
43 \( 1 + 438.T + 7.95e4T^{2} \)
47 \( 1 - 31.9iT - 1.03e5T^{2} \)
53 \( 1 - 2.84T + 1.48e5T^{2} \)
59 \( 1 + 71.6iT - 2.05e5T^{2} \)
61 \( 1 + 920.T + 2.26e5T^{2} \)
67 \( 1 + 444. iT - 3.00e5T^{2} \)
71 \( 1 + 541. iT - 3.57e5T^{2} \)
73 \( 1 + 764. iT - 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 - 603. iT - 5.71e5T^{2} \)
89 \( 1 - 1.15e3iT - 7.04e5T^{2} \)
97 \( 1 - 583. iT - 9.12e5T^{2} \)
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

−11.99917012494558840554970409748, −11.09583098268501163181796852144, −10.62688813572856950483510598410, −8.831849844537319841142278654433, −7.920226914223943102488097661154, −6.71988994509659037007533424365, −6.01141158397327719262452342127, −4.92080647819614805430536263350, −3.06668398118995282285012758210, −0.56230518469512623000815318040, 1.63745715889203622911525419577, 2.93807172633681479635222864994, 4.66180646825720732176447676806, 5.92483654729245152248346460982, 6.99196984756408918993086146917, 8.477475876693701524079402665456, 9.851320095427129331167715907163, 10.43086410591953671404894832238, 11.74366795104914102633916045856, 12.13729532696455875678303390095

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