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Properties

Label 1-117-117.41-r0-0-0
Degree $1$
Conductor $117$
Sign $0.622 - 0.782i$
Analytic cond. $0.543345$
Root an. cond. $0.543345$
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 + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·7-s − i·8-s + (0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s − i·20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·7-s − i·8-s + (0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s − i·20-s + (−0.5 + 0.866i)22-s + 23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.604231413 - 0.7734039433i\)
\(L(\frac12)\) \(\approx\) \(1.604231413 - 0.7734039433i\)
\(L(1)\) \(\approx\) \(1.600732572 - 0.5449102073i\)
\(L(1)\) \(\approx\) \(1.600732572 - 0.5449102073i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 - iT \)
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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

−29.61446404171438462919158639486, −28.74064379491257289164861193431, −26.833212014406589003925581149671, −26.16666715714569107499354717016, −25.32759920570841303801268198460, −24.062764576898129816555183235468, −23.36085600545329828831767837845, −22.22882208920345910107631563546, −21.34439896339080497830610641925, −20.526141937302554665826138992407, −19.045452666393559533191066918756, −17.51136345347555876408041539792, −16.95891714871517381352784072870, −15.58128218333763893846602117207, −14.554450009139120703362825282374, −13.47107197832292566712073931860, −12.99175310963429689295422502416, −11.14315669063071989686254267313, −10.36490211295830307370638804047, −8.55172082989936955680831414953, −7.193773715783785287115729994427, −6.27081863129125770087246118457, −5.03602798914912362905641343332, −3.63054907184681890804742653033, −2.25972455153924698819690018372, 1.82418953156203547298359647524, 2.85544073221777035136445673166, 4.800019279148252502219675234578, 5.50332759722377085569237611894, 6.786746342901429578632615795305, 8.76270936778531830157973248506, 9.85350589955274014383070324239, 11.01582557776060298760656901349, 12.4372519599479552789295227736, 12.971988020260812028249382069713, 14.20445369987146789656761691883, 15.27232913112716384041797535754, 16.29764099796566548939055608034, 17.8830886961454675944715680550, 18.78427324791183032262943889010, 20.16398017791047592489710336426, 21.07886151584688145712798711150, 21.71291237593443866029352080099, 22.82149795599708253556119145013, 23.902601727105448487061525552473, 25.02007913633563839789435802067, 25.51217667873973998365174662996, 27.3678391529679824756202578381, 28.54975364493679919061405037029, 28.947159645708128981620101159067

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