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Properties

Label 1-68-68.43-r1-0-0
Degree $1$
Conductor $68$
Sign $-0.739 + 0.673i$
Analytic cond. $7.30761$
Root an. cond. $7.30761$
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  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s + i·9-s + (−0.707 + 0.707i)11-s − 13-s − i·15-s − i·19-s − 21-s + (−0.707 + 0.707i)23-s + i·25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s + 33-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s + i·9-s + (−0.707 + 0.707i)11-s − 13-s − i·15-s − i·19-s − 21-s + (−0.707 + 0.707i)23-s + i·25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.707 − 0.707i)31-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(7.30761\)
Root analytic conductor: \(7.30761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (1:\ ),\ -0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03540574861 - 0.09143761274i\)
\(L(\frac12)\) \(\approx\) \(-0.03540574861 - 0.09143761274i\)
\(L(1)\) \(\approx\) \(0.5485759462 - 0.1978674144i\)
\(L(1)\) \(\approx\) \(0.5485759462 - 0.1978674144i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 - T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−32.23537414737460960995465416419, −31.3853854529131144569263002526, −30.14836459914646785538471443552, −28.921598187173914008984083406534, −27.84170685711695220554663264707, −26.98213953984911928394537426784, −26.16280119673551783011670852469, −24.322907531904720024453597771232, −23.503317031413785729220931522191, −22.11611142942115655775675836258, −21.63921443112631065991452783389, −20.14721000676010403610063736071, −18.66694697295955363670609948443, −17.78049579475096508012604612099, −16.32518605082529250116015543939, −15.329002050660999446359576706773, −14.45440616996498257355063569625, −12.37359142822900617038306778984, −11.329225557252949699044368840182, −10.50042334484689541717290069638, −8.86776216554318716837367842718, −7.3508368688328322297814200335, −5.730223482305987972952203420575, −4.51102580357978981783886397692, −2.83233283449148945465403728539, 0.053458744768271826279583459391, 1.75626202507227619670843841430, 4.32396927166559750922552711813, 5.44175330725165683489446420470, 7.39046900194963518837422699546, 7.95266557171946046023872671275, 10.05076776037061433788208274884, 11.45644911606873751090084462384, 12.35502386319734975857542231140, 13.46533275970566284340557860617, 15.03667428485329990958168522448, 16.53508496495308508473974367456, 17.31746693238317996169175200698, 18.51535729395554815581572582910, 19.80893951255029823399684095531, 20.75327592964288218402632929768, 22.38319604922497078515699556935, 23.583241383771285596374288809864, 24.01144486927752400468342305684, 25.21681171891231242194660870751, 26.90840391096885003812091975388, 27.79615179481607454870478692720, 28.81787735957541414581169057007, 29.870263383332479321332759050501, 30.9318126371679797403116758565

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