L-function 1-119-119.26-r1-0-0

• Label: 1-119-119.26-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $0.823 - 0.567i$
• Analytic cond.: $12.7883$
• Root an. cond.: $12.7883$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 + 0.5i)2^-s + (0.965 − 0.258i)3^-s + (0.5 − 0.866i)4^-s + (0.258 − 0.965i)5^-s + (−0.707 + 0.707i)6^-s + i·8^-s + (0.866 − 0.5i)9^-s + (0.258 + 0.965i)10^-s + (0.258 + 0.965i)11^-s + (0.258 − 0.965i)12^-s + 13^-s − i·15^-s + (−0.5 − 0.866i)16^-s + (−0.5 + 0.866i)18^-s + (0.866 − 0.5i)19^-s + (−0.707 − 0.707i)20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$0.823 - 0.567i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (26, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ 0.823 - 0.567i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.742604228 - 0.5426986170i$
$L(1)$	$\approx$	$1.158163650 - 0.1229890846i$

Euler product

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

	$p$	$F_p(T)$
bad	7	$1$
	17	$1$
good	2	$1 + (-0.866 + 0.5i)T$
	3	$1 + (0.965 - 0.258i)T$
	5	$1 + (0.258 - 0.965i)T$
	11	$1 + (0.258 + 0.965i)T$
	13	$1 + T$
	19	$1 + (0.866 - 0.5i)T$
	23	$1 + (-0.965 - 0.258i)T$
	29	$1 + (-0.707 - 0.707i)T$
	31	$1 + (0.965 - 0.258i)T$

Imaginary part of the first few zeros on the critical line

−29.05683416708101414879140632176, −27.68373666904112009383085830827, −26.864577333767311396797385076810, −26.09677433364249453426034252446, −25.41361205615620824551227383722, −24.29612592572330872385305556358, −22.43361263918164211000025665997, −21.55515488321381261506628593315, −20.685683705463283756934194326064, −19.61129279288611333435981042518, −18.713047589595681339819849651165, −18.05737448624892945675856055828, −16.45928457839909620972355376204, −15.56227258013845426724161855115, −14.18950872636208655179794483169, −13.338379643827752468877906888923, −11.63192183149556118562636854858, −10.60067927000352148767843796516, −9.67826443135784974715209829096, −8.55819941725764013666266654784, −7.58545852767520693153844819996, −6.23872041428022993217145742555, −3.72372471074241880436607649640, −2.97116224562166065620436078789, −1.51299155529510409697429112992, 1.02326237179548332211040915661, 2.19728884454937633674536023303, 4.2794500487840304225961745058, 5.90711737995345550190364975166, 7.294680907787454601599661758292, 8.30773635960411712217260558454, 9.22252914599826590001802297623, 10.0109209717788535413748001334, 11.7989831806906156594778802331, 13.1636849946900450368893978221, 14.18438053967815148293500071006, 15.41394026142249678717602713859, 16.21065724945120332049321943127, 17.55629203192261212342071334519, 18.32840426973165559676740847788, 19.636121914439396093099330868354, 20.30553809506669602536583956965, 21.09459812520991898002134762681, 23.043415390649262001486815100825, 24.27816358443805910518689490301, 24.80819963871329093790263272115, 25.79251445496052224480347510581, 26.45462576672177833498239891793, 27.89636378256764286759041738221, 28.37270938058431194912861976291

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