L-function 1-156-156.83-r1-0-0

• Label: 1-156-156.83-r1-0-0
• Degree: $1$
• Conductor: $156$
• Sign: $-0.289 + 0.957i$
• Analytic cond.: $16.7645$
• Root an. cond.: $16.7645$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·5^-s − i·7^-s + i·11^-s + 17^-s + i·19^-s − 23^-s − 25^-s − 29^-s + i·31^-s + 35^-s + i·37^-s + i·41^-s + 43^-s + i·47^-s − 49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign:	$-0.289 + 0.957i$
Analytic conductor:	\(16.7645\)
Root analytic conductor:	\(16.7645\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{156} (83, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 156,\ (1:\ ),\ -0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7905489547 + 1.065349498i\)
\(L(1)\)	\(\approx\)	\(0.9629445133 + 0.2915561391i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	13	\( 1 \)
good	5	\( 1 \)
	7	\( 1 \)
	11	\( 1 \)
	17	\( 1 \)
	19	\( 1 - iT \)
	23	\( 1 \)
	29	\( 1 \)
	31	\( 1 \)
	37	\( 1 + iT \)

Imaginary part of the first few zeros on the critical line

−27.77562987153526294087777853142, −26.36003295749242935448144052867, −25.334288116201826993633866102645, −24.380660703840901600526181669065, −23.81788599512921900410089217029, −22.316752708775455113140928503232, −21.47390024874872394672942764364, −20.61471235760106561016425434706, −19.42892780387700475110256073061, −18.57301916997727361300071338855, −17.34830985058547583892854955872, −16.31674645082879068141207859943, −15.55532578949516553595397888065, −14.230082374996387668715940077539, −13.08632629856897314912676918400, −12.15965601682234077438328471713, −11.207364643840820815153053628241, −9.59308248953805830577693708282, −8.77047962755021680381268702727, −7.78389717459136064339750221154, −5.97650688912942371433505584158, −5.24847680067723015228663918431, −3.72932571653588901004922412823, −2.157020363890858921805921681585, −0.498159412106850487793084297622, 1.59910775196490900489612011329, 3.21631633378325478307967088815, 4.320022370523815341672545757623, 5.96278386935602555327514200489, 7.16377641453551323797689012228, 7.89207187406111397493493820308, 9.82220747076534186215505132982, 10.34991617817862654786516662247, 11.58598707266241941439331337782, 12.767811652812311811527990662603, 14.12644367150559723128764425694, 14.65461410400428289673673393932, 15.97722096760541939781661008813, 17.11105978464597943396769031542, 18.05818060880996978256683002461, 19.026510287051759937863753406924, 20.122122683751825305711990038561, 21.00657954608994555208235565375, 22.33274110371869209560765669939, 23.02755941793886884233970574127, 23.83560862153370405992837407663, 25.348264954989370445275522188159, 25.99019119690539952334912893419, 26.96030373026663149207982039523, 27.79564157912691555825504718399

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