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Properties

Label 1-208-208.115-r0-0-0
Degree $1$
Conductor $208$
Sign $0.577 + 0.816i$
Analytic cond. $0.965947$
Root an. cond. $0.965947$
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)3-s + 5-s + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + i·27-s + (−0.866 − 0.5i)29-s − i·31-s + (−0.866 + 0.5i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + 5-s + (−0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)23-s + 25-s + i·27-s + (−0.866 − 0.5i)29-s − i·31-s + (−0.866 + 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(0.965947\)
Root analytic conductor: \(0.965947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 208,\ (0:\ ),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451249113 + 0.7508492278i\)
\(L(\frac12)\) \(\approx\) \(1.451249113 + 0.7508492278i\)
\(L(1)\) \(\approx\) \(1.373181057 + 0.3962365477i\)
\(L(1)\) \(\approx\) \(1.373181057 + 0.3962365477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + T \)
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

−26.38381188178192097220401663137, −25.538286322476110998711914026673, −25.03284759415966830184878684148, −23.889157039823347775340318199522, −22.9640688906156914391981475356, −21.667970306987852317818307460235, −20.90132271154626962819937897625, −19.98978047144703643910326374747, −18.90491087603800648972178732226, −18.3182924065932813214246699991, −17.010655082638222020370170045906, −16.120584797303624799992718261197, −14.788067293746134016966193368929, −13.72585891526056683427628846354, −13.34268234034413990024825788784, −12.28063467858079143417156112965, −10.61022250337768062924025825203, −9.67604518478393752900403464760, −8.82299645047461310683994068454, −7.54265083301728307239369898925, −6.543104567112267892573789355048, −5.45371847084993283228647488296, −3.58302789317333015850506948745, −2.71481616500105248328635752411, −1.267204670644458643291941617189, 2.05941203902467678439138759502, 2.850252275805990470762694563985, 4.31485273923772197948497789633, 5.56793310270852380537025401188, 6.75536486496989682686697814524, 8.13315631928765999097490547261, 9.33182966324931061639154081665, 9.81412292084414361573834043450, 10.866974078720239851827681805095, 12.879698329573461719257077006186, 13.04027857082346431855520128768, 14.52508429311468184872304834170, 15.15439952892516165760105277377, 16.256253670685467177701583371405, 17.26716398515072424207973646915, 18.490224063221264982296751935339, 19.32724903968104794923987634545, 20.406043300763377233269401943297, 21.22739646593831111658306820007, 21.976881981320193415644763701069, 22.90782687014035179289788851428, 24.39311835559872459098087867531, 25.296643312628878759068505192359, 25.93767969735012642664067306407, 26.45168342023895578623591267986

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