L-function 1-119-119.104-r1-0-0

• Label: 1-119-119.104-r1-0-0
• Degree: $1$
• Conductor: $119$
• Sign: $0.0465 + 0.998i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$119$    =    $7 \cdot 17$
Sign:	$0.0465 + 0.998i$
Analytic conductor:	$12.7883$
Root analytic conductor:	$12.7883$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{119} (104, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 119,\ (1:\ ),\ 0.0465 + 0.998i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.066469631 + 1.017966179i$
$L(1)$	$\approx$	$0.8404659796 + 0.5374673593i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−28.92167060237542181435664995847, −27.89883046894129371580915458288, −26.88677282860871406256465082334, −25.63726567718300948646105282593, −24.47970725280634799134514195143, −23.150565705732241763443801647400, −22.48703979382474311162939262044, −21.607683555952599830581238569615, −20.509578656241156500198515585323, −18.8992602453156359741840638794, −18.67813161460662234422070960688, −17.51502821024931154992006284961, −16.61655532565044400706742325637, −14.52150374107314659716874412424, −13.61323190702275513947841390927, −12.73521229464190250832872625854, −11.37802795597352980230999150356, −10.85327752319942792056246588287, −9.55022063816824055719577028055, −8.12925354977670364495999194388, −6.44353938097737956140880596542, −5.55447996823304017732847671032, −3.73733941075828220862983708265, −2.21451011697088777873194813022, −0.96920045969048869322380202660, 1.028427592119508514596104409711, 3.90620393584661786679158677310, 5.00753774132285936283521415677, 5.914855099386449715940827932400, 7.04591050183741815853336002780, 8.95000652481262482695264314073, 9.3989389995519763351271608825, 10.828200517957034466265427125414, 12.39865344190506494682822716241, 13.44373372343915097522536364191, 14.73107438311144787378516150315, 15.762622746942457547177367091492, 16.66848000356350032117414239748, 17.45656502779304948950889402518, 18.21171388784260202152695717708, 20.04673561725111237127504267326, 21.304181340290929977875469268112, 22.08749195340224838046977675629, 23.163835450410543026202007188188, 23.97547001767572985064524104329, 25.22097127843420203348670698344, 25.893117046149125901352625623762, 27.16904219248149258265353660437, 28.04056687804854903097894452865, 28.64052041436982823655320099563

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