L-function 10-871e5-871.870-c0e5-0-0

• Label: 10-871e5-871.870-c0e5-0-0
• Degree: $10$
• Conductor: $5.013\times 10^{14}$
• Sign: $1$
• Analytic cond.: $0.0155194$
• Root an. cond.: $0.659306$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: no
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 2^-s − 5^-s − 7^-s + 5·9^-s + 10^-s − 11^-s + 5·13^-s + 14^-s − 17^-s − 5·18^-s + 22^-s − 23^-s − 5·26^-s − 29^-s − 31^-s + 34^-s + 35^-s − 41^-s − 5·45^-s + 46^-s + 55^-s + 58^-s + 62^-s − 5·63^-s − 5·65^-s + 5·67^-s − 70^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}$

Invariants

Degree:	$10$
Conductor:	$13^{5} \cdot 67^{5}$
Sign:	$1$
Analytic conductor:	$0.0155194$
Root analytic conductor:	$0.659306$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	induced by $\chi_{871} (870, \cdot )$
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(10,\ 13^{5} \cdot 67^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5050803489$
$L(1)$		not available

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	13	$C_1$	$( 1 - T )^{5}$
	67	$C_1$	$( 1 - T )^{5}$
good	2	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	3	$C_1$$\times$$C_1$	$( 1 - T )^{5}( 1 + T )^{5}$
	5	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	7	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	11	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	17	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	19	$C_1$$\times$$C_1$	$( 1 - T )^{5}( 1 + T )^{5}$
	23	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$
	29	$C_{10}$	$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$

Imaginary part of the first few zeros on the critical line

−6.41521777442232251601523839744, −6.38007336214568427242312476651, −6.22579924089976122148137893873, −5.96606895551648770807377357818, −5.67699795498372690253040182018, −5.51700652968924302243666686000, −5.06816699709763172518015467504, −5.02011044839378537216726805350, −4.99486295131559950778141150225, −4.49517242692864748899786132279, −4.12663462674993278134103642721, −4.06186531403159952222565163371, −4.02487493966216550249959192141, −3.93534756426053421465410465936, −3.65639697030646127513698771091, −3.55612899544479759184163460845, −3.41765566975277048208790339315, −3.10648689350912656989694630407, −2.45630345460003225869083086521, −2.21769598878851124827386249683, −2.00370362099285550622550028097, −1.55079578150419719949774376287, −1.48755455653452536025097078877, −1.16451097523120678313979026605, −0.913512058416947344120449872157, 0.913512058416947344120449872157, 1.16451097523120678313979026605, 1.48755455653452536025097078877, 1.55079578150419719949774376287, 2.00370362099285550622550028097, 2.21769598878851124827386249683, 2.45630345460003225869083086521, 3.10648689350912656989694630407, 3.41765566975277048208790339315, 3.55612899544479759184163460845, 3.65639697030646127513698771091, 3.93534756426053421465410465936, 4.02487493966216550249959192141, 4.06186531403159952222565163371, 4.12663462674993278134103642721, 4.49517242692864748899786132279, 4.99486295131559950778141150225, 5.02011044839378537216726805350, 5.06816699709763172518015467504, 5.51700652968924302243666686000, 5.67699795498372690253040182018, 5.96606895551648770807377357818, 6.22579924089976122148137893873, 6.38007336214568427242312476651, 6.41521777442232251601523839744

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