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Properties

Label 1-119-119.26-r1-0-0
Degree $1$
Conductor $119$
Sign $0.823 - 0.567i$
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  + (−0.866 + 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 − 0.866i)4-s + (0.258 − 0.965i)5-s + (−0.707 + 0.707i)6-s + i·8-s + (0.866 − 0.5i)9-s + (0.258 + 0.965i)10-s + (0.258 + 0.965i)11-s + (0.258 − 0.965i)12-s + 13-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + (−0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 − 0.866i)4-s + (0.258 − 0.965i)5-s + (−0.707 + 0.707i)6-s + i·8-s + (0.866 − 0.5i)9-s + (0.258 + 0.965i)10-s + (0.258 + 0.965i)11-s + (0.258 − 0.965i)12-s + 13-s − i·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + (−0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.823 - 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.742604228 - 0.5426986170i\)
\(L(\frac12)\) \(\approx\) \(1.742604228 - 0.5426986170i\)
\(L(1)\) \(\approx\) \(1.158163650 - 0.1229890846i\)
\(L(1)\) \(\approx\) \(1.158163650 - 0.1229890846i\)

Euler product

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

−29.05683416708101414879140632176, −27.68373666904112009383085830827, −26.864577333767311396797385076810, −26.09677433364249453426034252446, −25.41361205615620824551227383722, −24.29612592572330872385305556358, −22.43361263918164211000025665997, −21.55515488321381261506628593315, −20.685683705463283756934194326064, −19.61129279288611333435981042518, −18.713047589595681339819849651165, −18.05737448624892945675856055828, −16.45928457839909620972355376204, −15.56227258013845426724161855115, −14.18950872636208655179794483169, −13.338379643827752468877906888923, −11.63192183149556118562636854858, −10.60067927000352148767843796516, −9.67826443135784974715209829096, −8.55819941725764013666266654784, −7.58545852767520693153844819996, −6.23872041428022993217145742555, −3.72372471074241880436607649640, −2.97116224562166065620436078789, −1.51299155529510409697429112992, 1.02326237179548332211040915661, 2.19728884454937633674536023303, 4.2794500487840304225961745058, 5.90711737995345550190364975166, 7.294680907787454601599661758292, 8.30773635960411712217260558454, 9.22252914599826590001802297623, 10.0109209717788535413748001334, 11.7989831806906156594778802331, 13.1636849946900450368893978221, 14.18438053967815148293500071006, 15.41394026142249678717602713859, 16.21065724945120332049321943127, 17.55629203192261212342071334519, 18.32840426973165559676740847788, 19.636121914439396093099330868354, 20.30553809506669602536583956965, 21.09459812520991898002134762681, 23.043415390649262001486815100825, 24.27816358443805910518689490301, 24.80819963871329093790263272115, 25.79251445496052224480347510581, 26.45462576672177833498239891793, 27.89636378256764286759041738221, 28.37270938058431194912861976291

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