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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(1.066469631 + 1.017966179i\)
\(L(\frac12)\) \(\approx\) \(1.066469631 + 1.017966179i\)
\(L(1)\) \(\approx\) \(0.8404659796 + 0.5374673593i\)
\(L(1)\) \(\approx\) \(0.8404659796 + 0.5374673593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + iT \)
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 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 + iT \)
59 \( 1 + iT \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + T \)
97 \( 1 + (0.707 - 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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