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Properties

Label 1-119-119.109-r1-0-0
Degree $1$
Conductor $119$
Sign $0.100 - 0.994i$
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  + (0.965 − 0.258i)2-s + (−0.608 + 0.793i)3-s + (0.866 − 0.5i)4-s + (0.130 − 0.991i)5-s + (−0.382 + 0.923i)6-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.130 − 0.991i)10-s + (−0.991 + 0.130i)11-s + (−0.130 + 0.991i)12-s − i·13-s + (0.707 + 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.965 − 0.258i)19-s + (−0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.608 + 0.793i)3-s + (0.866 − 0.5i)4-s + (0.130 − 0.991i)5-s + (−0.382 + 0.923i)6-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.130 − 0.991i)10-s + (−0.991 + 0.130i)11-s + (−0.130 + 0.991i)12-s − i·13-s + (0.707 + 0.707i)15-s + (0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (0.965 − 0.258i)19-s + (−0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.720098173 - 1.554331372i\)
\(L(\frac12)\) \(\approx\) \(1.720098173 - 1.554331372i\)
\(L(1)\) \(\approx\) \(1.472723223 - 0.4767351903i\)
\(L(1)\) \(\approx\) \(1.472723223 - 0.4767351903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
5 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (-0.991 + 0.130i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 + (0.608 - 0.793i)T \)
37 \( 1 + (0.991 + 0.130i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.793 + 0.608i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.793 + 0.608i)T \)
79 \( 1 + (0.608 + 0.793i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−29.16801669056498336048110030468, −28.74100205799734592693196778714, −26.82311942914898094086080311872, −25.8255183088102727668875981294, −24.88983979338560899174785463331, −23.6677531217015613348495772440, −23.27716899242652039122050965864, −22.08670686925369609563848906490, −21.43716015894558612311397081563, −19.872275793941314999800879647946, −18.67196365114659127681150016036, −17.792771839345826223669749696149, −16.50098863609834257656982838410, −15.508921571637078432105064908845, −14.05736632565921326283491823856, −13.569351069199547294765031071461, −12.17172442922332550066204704394, −11.37394296190668425562450452014, −10.29702087520379520656293763639, −7.95199792305535054846853346973, −7.02145226324608095830103707247, −6.10951113217221445928760850807, −4.98179373466007111977717214801, −3.19360326635810246796171950153, −1.931439818847609255593539439530, 0.746558556289842788367235680184, 2.835170432845601337968433312570, 4.36957418383459697669089614833, 5.1891613699808632607166857857, 6.09242850834953851671974829246, 7.9624232658125126299746127164, 9.699389679827234078275723213117, 10.5546339809542895853550317341, 11.82541420987066567382354352241, 12.68177643484710917579464909065, 13.749530372918222962177706333539, 15.293435271708766231903762225472, 15.88030167644824535526966746333, 16.90493327538904391962999617585, 18.187502992823092690430523949528, 20.07218763957322466833231035318, 20.593170379761345576461541554799, 21.51493053765241007650049030098, 22.49189013317486573003052569742, 23.410528388434516398352204572357, 24.289953302921034107166734788748, 25.377166115023802524839911866700, 26.75704266686276185609780502718, 28.11203849855328271827319270797, 28.58018145914246863004607892997

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