L-function 1-152-152.37-r1-0-0

• Label: 1-152-152.37-r1-0-0
• Degree: $1$
• Conductor: $152$
• Sign: $1$
• Analytic cond.: $16.3346$
• Root an. cond.: $16.3346$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 3^-s − 5^-s + 7^-s + 9^-s − 11^-s + 13^-s − 15^-s + 17^-s + 21^-s + 23^-s + 25^-s + 27^-s + 29^-s − 31^-s − 33^-s − 35^-s + 37^-s + 39^-s − 41^-s − 43^-s − 45^-s + 47^-s + 49^-s + 51^-s + 53^-s + 55^-s + 59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$152$    =    $2^{3} \cdot 19$
Sign:	$1$
Analytic conductor:	$16.3346$
Root analytic conductor:	$16.3346$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{152} (37, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(1,\ 152,\ (1:\ ),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.547011170$
$L(1)$	$\approx$	$1.528900874$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	19	$1$
good	3	$1 + T$
	5	$1 - T$
	7	$1 + T$
	11	$1 - T$
	13	$1 + T$
	17	$1 + T$
	23	$1 + T$
	29	$1 + T$
	31	$1 - T$

Imaginary part of the first few zeros on the critical line

−27.45887285573937389820892798541, −26.94809649814633933966714367866, −25.85024417749752603888994601316, −24.95406852489626898469612227179, −23.72252832878027149694686841421, −23.335080669753099723586315482, −21.51870268947889990682587448961, −20.77525963658662156807406933432, −20.01976512568988203097134503495, −18.77260169016316670869643134353, −18.2645953095371647299663582579, −16.52909859818801048029930065168, −15.4792864110982358339660426088, −14.79826939888833507246297202049, −13.69400838589107345023346018473, −12.5982080877174889675810617196, −11.331986857680979264308302444429, −10.31849529825159021449394279600, −8.69716745995560468494862953100, −8.06630160341600837143926917835, −7.174690552612637897130104196306, −5.17808792097857683595651995091, −3.97177712649849110830621889787, −2.84505871867905888360710213691, −1.191133512716638333409087237597, 1.191133512716638333409087237597, 2.84505871867905888360710213691, 3.97177712649849110830621889787, 5.17808792097857683595651995091, 7.174690552612637897130104196306, 8.06630160341600837143926917835, 8.69716745995560468494862953100, 10.31849529825159021449394279600, 11.331986857680979264308302444429, 12.5982080877174889675810617196, 13.69400838589107345023346018473, 14.79826939888833507246297202049, 15.4792864110982358339660426088, 16.52909859818801048029930065168, 18.2645953095371647299663582579, 18.77260169016316670869643134353, 20.01976512568988203097134503495, 20.77525963658662156807406933432, 21.51870268947889990682587448961, 23.335080669753099723586315482, 23.72252832878027149694686841421, 24.95406852489626898469612227179, 25.85024417749752603888994601316, 26.94809649814633933966714367866, 27.45887285573937389820892798541

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