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Properties

Label 1-119-119.76-r1-0-0
Degree $1$
Conductor $119$
Sign $0.651 + 0.758i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.135755288 + 0.5218143797i\)
\(L(\frac12)\) \(\approx\) \(1.135755288 + 0.5218143797i\)
\(L(1)\) \(\approx\) \(0.9932238307 - 0.09359833009i\)
\(L(1)\) \(\approx\) \(0.9932238307 - 0.09359833009i\)

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

−28.64433396988932556632155792289, −27.336908593739767944047964199671, −26.22341138657156456383294772334, −25.98475923035810596160657071524, −24.60279698034473549908908563053, −23.79273809711065619784140727936, −23.09422731759727126424587714978, −21.83130007927972140865292267242, −20.38261538572360128055243298082, −18.94330623958736069036622134049, −18.62634225689534810057664669092, −17.467893443387885269463662373063, −15.83420321408948978919961847828, −15.30081995949708908260097208636, −13.9861526954092973878607607281, −13.415306370844265167951052229935, −11.974820083681106320227488323575, −10.42241361762782070134684969991, −8.73562967950483095907803400391, −8.057380217723522164158218165345, −6.98269282289367203548235157457, −6.0191293854607870098673122341, −4.11055424211149688621831264538, −2.874587957609983150099830440637, −0.49785906472684130237664124453, 1.60008466253384032841284463647, 3.24084664838334170701847258916, 4.20932430876842882094987195398, 5.25672625670749015967397115052, 7.89458258983629590576236047441, 8.63318260348742417113767746297, 9.80830502609424938338731209980, 10.77053000533154567514106759448, 12.048525484385350562680303794020, 13.09303263768420052062635832511, 14.158871793049788468252911010549, 15.46141038993395969892461805439, 16.36626118532114004315590373310, 17.89993789764907162855620752133, 19.10803206430966654838558879249, 20.02834613336583186987952265540, 20.726668859223237417411902534695, 21.4384825772911398355217734657, 22.84024094956311730630829758091, 23.60307155346922155614322195010, 25.20206872928969940930653485239, 26.26497820012434772001482140348, 27.25386153825374280948573073362, 28.00519224216009894582768495069, 28.72377664733637559952132066792

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