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Properties

Label 1-119-119.44-r1-0-0
Degree $1$
Conductor $119$
Sign $-0.571 + 0.820i$
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  + (−0.258 + 0.965i)2-s + (−0.991 − 0.130i)3-s + (−0.866 − 0.5i)4-s + (−0.793 − 0.608i)5-s + (0.382 − 0.923i)6-s + (0.707 − 0.707i)8-s + (0.965 + 0.258i)9-s + (0.793 − 0.608i)10-s + (−0.608 − 0.793i)11-s + (0.793 + 0.608i)12-s − i·13-s + (0.707 + 0.707i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + (0.382 + 0.923i)20-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.991 − 0.130i)3-s + (−0.866 − 0.5i)4-s + (−0.793 − 0.608i)5-s + (0.382 − 0.923i)6-s + (0.707 − 0.707i)8-s + (0.965 + 0.258i)9-s + (0.793 − 0.608i)10-s + (−0.608 − 0.793i)11-s + (0.793 + 0.608i)12-s − i·13-s + (0.707 + 0.707i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.258 + 0.965i)19-s + (0.382 + 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.571 + 0.820i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ -0.571 + 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1644254464 + 0.3148625131i\)
\(L(\frac12)\) \(\approx\) \(0.1644254464 + 0.3148625131i\)
\(L(1)\) \(\approx\) \(0.4650189270 + 0.1359421565i\)
\(L(1)\) \(\approx\) \(0.4650189270 + 0.1359421565i\)

Euler product

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

−28.29031823402423632501533073214, −27.95715934837085950999035740683, −26.649180790293509335201127789460, −26.10455464838368532680522496263, −23.97410749426300135961653522457, −23.21085166016812039400728822210, −22.34044538270230463328805226753, −21.497131838114882179292264595502, −20.31894187919470134993666699591, −19.112923292452403193143934432331, −18.331460889757701884346468917632, −17.4001598196116973518071140568, −16.19468226845691858022666329741, −15.01303243983288697482450115895, −13.42196216338123154008785257317, −12.17637204514843940301823253788, −11.49469738930996680996507927699, −10.56152587442808649752332060161, −9.56108922721041073252440467724, −7.89348538940792340344701127467, −6.697007154918785093544880204417, −4.84220365290720025356200580842, −3.924989094709502047466679400714, −2.192547472672918013567640305962, −0.252771943359995701759839104323, 0.92030512850127668572239321946, 3.96502127206630028030225791713, 5.24394295850894199559581625966, 6.060318284090750289010439019419, 7.59817435248188010751862089426, 8.29421999773471387393562997264, 9.93167655120651501716994220315, 11.065450308422779124970310076047, 12.45414723468738566569474206691, 13.33390690193681356436033258933, 14.9688835666417222450268817667, 16.07877186111510818574774878413, 16.500423668150510106506465961218, 17.73423996711551604500366235422, 18.59813428041866898825757787330, 19.68757308413743909529768626016, 21.261755079017611783456266924476, 22.62391317295463999304235769778, 23.25953932077222146269615645028, 24.20048048899237771378046367329, 24.81804781686123663027249934675, 26.339989793541997289871485618444, 27.36945829074999305464847625656, 27.913411584728702611452097771698, 28.9178198324297689065748804532

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