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Properties

Label 1-119-119.62-r0-0-0
Degree $1$
Conductor $119$
Sign $0.861 + 0.507i$
Analytic cond. $0.552633$
Root an. cond. $0.552633$
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  + (0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s + i·4-s + (−0.382 − 0.923i)5-s + (0.923 + 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)12-s + i·13-s + (−0.707 − 0.707i)15-s − 16-s + 18-s + (−0.707 − 0.707i)19-s + (0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s + i·4-s + (−0.382 − 0.923i)5-s + (0.923 + 0.382i)6-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)10-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)12-s + i·13-s + (−0.707 − 0.707i)15-s − 16-s + 18-s + (−0.707 − 0.707i)19-s + (0.923 − 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.765910286 + 0.4815188991i\)
\(L(\frac12)\) \(\approx\) \(1.765910286 + 0.4815188991i\)
\(L(1)\) \(\approx\) \(1.687636882 + 0.3747991664i\)
\(L(1)\) \(\approx\) \(1.687636882 + 0.3747991664i\)

Euler product

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

−29.6344452227879528874243087771, −27.59255988300040482071919755477, −27.459623092141940365670817578537, −26.006149575814785171624842571195, −24.99494722398406080694441082299, −23.85420639337678976851125542197, −22.53252655130143564915698173337, −22.00413845823367405733318673922, −20.85158337022764946204522293100, −19.816055833532831237360840713092, −19.21115966346576997893427743755, −18.11989687585955069274995951538, −16.127971208253039041881726632023, −14.95444024352980084872272139277, −14.47745794993315353634080447269, −13.40289411482189382321061318832, −12.11689067080045846328480981831, −10.82650269203945807874091448104, −10.06943307507726966287682840249, −8.70687698955744099028379859073, −7.22197810222478390418716418606, −5.74036371012322366394656260747, −3.97673212758748187320657588223, −3.355522922111378117712209034299, −2.0012387153678930156894085317, 1.98461238870562714170332537638, 3.789714061308348958920069395, 4.567340118318351366573841766968, 6.36168293930814709534056548844, 7.42376703814376158120229804293, 8.58026301592217581510958310677, 9.306603610906063105907721367837, 11.73670053880536101760213588662, 12.549573883970882423042605402802, 13.57075068264596927051752392265, 14.502490406175777494866262830554, 15.4928835825977283345561493670, 16.55585397826835460797466755575, 17.58501951014279092399806269193, 19.12869874303062979917914455636, 20.14797755824892199050031208162, 21.01112175678592466778712167958, 22.103355530555570142115695535478, 23.57638711609699274805835799130, 24.16039753406985055591253047837, 25.01964356211767680144251249158, 25.89154000806526464583186936836, 26.883525807350879831252852924107, 28.089503057852164011995294562012, 29.58328276773424456376082524049

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