Properties

Label 4-13e2-1.1-c3e2-0-2
Degree $4$
Conductor $169$
Sign $1$
Analytic cond. $0.588327$
Root an. cond. $0.875799$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7·4-s − 51·9-s − 14·12-s + 52·13-s − 15·16-s − 90·17-s + 324·23-s + 169·25-s + 158·27-s − 288·29-s − 357·36-s − 104·39-s − 194·43-s + 30·48-s + 461·49-s + 180·51-s + 364·52-s − 828·53-s + 752·61-s − 553·64-s − 630·68-s − 648·69-s − 338·75-s − 1.66e3·79-s + 1.86e3·81-s + 576·87-s + ⋯
L(s)  = 1  − 0.384·3-s + 7/8·4-s − 1.88·9-s − 0.336·12-s + 1.10·13-s − 0.234·16-s − 1.28·17-s + 2.93·23-s + 1.35·25-s + 1.12·27-s − 1.84·29-s − 1.65·36-s − 0.427·39-s − 0.688·43-s + 0.0902·48-s + 1.34·49-s + 0.494·51-s + 0.970·52-s − 2.14·53-s + 1.57·61-s − 1.08·64-s − 1.12·68-s − 1.13·69-s − 0.520·75-s − 2.36·79-s + 2.56·81-s + 0.709·87-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $1$
Analytic conductor: \(0.588327\)
Root analytic conductor: \(0.875799\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{13} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 169,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8760006070\)
\(L(\frac12)\) \(\approx\) \(0.8760006070\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ \( 1 - 4 p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - 7 T^{2} + p^{6} T^{4} \)
3$C_2$ \( ( 1 + T + p^{3} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 169 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 461 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 358 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 45 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 13682 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 162 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 144 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10114 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 9497 T^{2} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 100978 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 + 97 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 195325 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 414 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 138274 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 376 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 600230 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 588373 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 592 T + p^{3} T^{2} )( 1 + 592 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 + 830 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 951730 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 1218094 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 1099442 T^{2} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.07747087618661690760159125164, −19.12066855542506934627592696216, −18.56570016008740859489133774140, −17.62314566303340338455370062924, −17.00472259346844510464561559806, −16.63463237935173972028166248689, −15.74602187090146680512445425353, −15.05289148393711810595773703217, −14.46122281821318661474247024287, −13.40313563714491926733499154473, −12.82368120640982122209391662241, −11.51890229060460885251302265496, −11.04973796206346326445617763542, −10.96032656074757330025078934392, −8.983836264546810122151905930528, −8.704185250031597270937396060796, −7.09714183483057059409876765045, −6.30719010199337973804419825585, −5.18901415598640781945830316564, −2.95310855429547515318810034846, 2.95310855429547515318810034846, 5.18901415598640781945830316564, 6.30719010199337973804419825585, 7.09714183483057059409876765045, 8.704185250031597270937396060796, 8.983836264546810122151905930528, 10.96032656074757330025078934392, 11.04973796206346326445617763542, 11.51890229060460885251302265496, 12.82368120640982122209391662241, 13.40313563714491926733499154473, 14.46122281821318661474247024287, 15.05289148393711810595773703217, 15.74602187090146680512445425353, 16.63463237935173972028166248689, 17.00472259346844510464561559806, 17.62314566303340338455370062924, 18.56570016008740859489133774140, 19.12066855542506934627592696216, 20.07747087618661690760159125164

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