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Properties

Label 1-208-208.67-r0-0-0
Degree $1$
Conductor $208$
Sign $0.933 + 0.358i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.933 + 0.358i$
Analytic conductor: \(0.965947\)
Root analytic conductor: \(0.965947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 208,\ (0:\ ),\ 0.933 + 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473642487 + 0.2734206286i\)
\(L(\frac12)\) \(\approx\) \(1.473642487 + 0.2734206286i\)
\(L(1)\) \(\approx\) \(1.312025872 + 0.1625341757i\)
\(L(1)\) \(\approx\) \(1.312025872 + 0.1625341757i\)

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

−26.740328073148677814893691337656, −25.63150200433105190174108461430, −24.79308031489416752316028947154, −24.010488568757317619959770712044, −23.155730250869014544209226063246, −21.96290879034728154163062337632, −20.64118484022236122309283408756, −20.160720290541211474843254347839, −19.09384828758243440747726053536, −18.336996422997700050258445958053, −17.33393667741245075564863398312, −15.780501972069122443151357018211, −15.02765078504698480804803546964, −14.317643964922201939677899032466, −13.06269774417626845492707377338, −11.97864719619874802578267030164, −11.35768964254955316038752331614, −9.57301293399844669171059629846, −8.721276822209829395358832024378, −7.59737721381831460022032767519, −7.07188138077561381592889312490, −5.17362186973962037591775042270, −3.98004889131215564616550175720, −2.76005253901677648162604786719, −1.39357199323967491225598423132, 1.489761368472757078826328402417, 3.314006211685818177714679920078, 3.99291619791028507787517559098, 5.174781705012685271907684946247, 6.99495146163070284685585463469, 8.180701525156024931914241728026, 8.53845516844223636098066064971, 10.12235651180141113019404238876, 11.01878172026328277420914015003, 12.04371776832459346902985412413, 13.44072334079812178415696587352, 14.48840801877489717110128298507, 14.997175170930423676670181278894, 16.2531128294460942009266106245, 16.92388137682119467873973339750, 18.58691299195106091727107254681, 19.29190688261423185769790547693, 20.256697883855280699798945367228, 20.936317442250609416215342123831, 21.96483466627053431741480699929, 23.09625334851315354490425234964, 24.15989675118806652856426994112, 24.79078605081304484746729730572, 26.1007724408806362478738123475, 27.02887838432749814391080849414

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