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Properties

Label 1-117-117.2-r0-0-0
Degree $1$
Conductor $117$
Sign $0.317 - 0.948i$
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  + i·2-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)10-s − i·11-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)10-s − i·11-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.866 + 0.5i)20-s + 22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3292450601 - 0.2369250992i\)
\(L(\frac12)\) \(\approx\) \(0.3292450601 - 0.2369250992i\)
\(L(1)\) \(\approx\) \(0.6090937959 + 0.06529479632i\)
\(L(1)\) \(\approx\) \(0.6090937959 + 0.06529479632i\)

Euler product

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

−29.50795302803141454445524996912, −28.136879263180348129809657907770, −27.889554778763689250292327223821, −26.37098626152671863137548558154, −25.81581349715452826749431083839, −24.0017537566000330683784822692, −22.92972288393966281437580919913, −22.336705357629404378644250653951, −21.24073279590327027886937581891, −19.83652855885599002763346147546, −19.38921413565199612752871360146, −18.348578224782268285724031212004, −17.22830365917930970846062181445, −15.58735608161862453390411702585, −14.78721844506527420589786188508, −13.19642487033805769916435119021, −12.36872247756019260661545213814, −11.35643786704610314335005311349, −10.23911436423710475847353011343, −9.163038164444994351177446823139, −7.8526595688043139992204939257, −6.32569791510637181301570347426, −4.53494753158342548525103778894, −3.44492960355282019185458602976, −2.160234628339415276862694210143, 0.38069118126666308268950730210, 3.4705293195568457192849596705, 4.52300436544312309259483143704, 6.009470657551473163339220298907, 7.1355593493621804350102187278, 8.28641201881940150143618435352, 9.231424471748076478479927035962, 10.701722667706768515398271387024, 12.31385416348257562100854637202, 13.30109883732026163714525304847, 14.368325959587155290169153166537, 15.79759155543484759525656291613, 16.255396872202525486738529956779, 17.22715264617948457373772248361, 18.75603382391759357545601219612, 19.44104807794453291071130729847, 20.78549835306263448811109196368, 22.32511694143426623755567875255, 23.03264659636545802734086483810, 24.03987737627055843940116420088, 24.77099151608035788814501508665, 26.06990107452235571811770413665, 26.82994069130820118480252026764, 27.68162876017895986778738189198, 28.848144999699951678206963930005

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