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Properties

Label 1-26e2-676.123-r0-0-0
Degree $1$
Conductor $676$
Sign $-0.672 - 0.740i$
Analytic cond. $3.13933$
Root an. cond. $3.13933$
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  + (−0.692 − 0.721i)3-s + (0.935 + 0.354i)5-s + (0.600 − 0.799i)7-s + (−0.0402 + 0.999i)9-s + (−0.999 + 0.0402i)11-s + (−0.391 − 0.919i)15-s + (−0.799 − 0.600i)17-s + (−0.866 − 0.5i)19-s + (−0.992 + 0.120i)21-s + (−0.5 − 0.866i)23-s + (0.748 + 0.663i)25-s + (0.748 − 0.663i)27-s + (−0.845 − 0.534i)29-s + (−0.663 − 0.748i)31-s + (0.721 + 0.692i)33-s + ⋯
L(s)  = 1  + (−0.692 − 0.721i)3-s + (0.935 + 0.354i)5-s + (0.600 − 0.799i)7-s + (−0.0402 + 0.999i)9-s + (−0.999 + 0.0402i)11-s + (−0.391 − 0.919i)15-s + (−0.799 − 0.600i)17-s + (−0.866 − 0.5i)19-s + (−0.992 + 0.120i)21-s + (−0.5 − 0.866i)23-s + (0.748 + 0.663i)25-s + (0.748 − 0.663i)27-s + (−0.845 − 0.534i)29-s + (−0.663 − 0.748i)31-s + (0.721 + 0.692i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.672 - 0.740i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ -0.672 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3583050054 - 0.8094863878i\)
\(L(\frac12)\) \(\approx\) \(0.3583050054 - 0.8094863878i\)
\(L(1)\) \(\approx\) \(0.7997286349 - 0.3401817141i\)
\(L(1)\) \(\approx\) \(0.7997286349 - 0.3401817141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.692 - 0.721i)T \)
5 \( 1 + (0.935 + 0.354i)T \)
7 \( 1 + (0.600 - 0.799i)T \)
11 \( 1 + (-0.999 + 0.0402i)T \)
17 \( 1 + (-0.799 - 0.600i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.845 - 0.534i)T \)
31 \( 1 + (-0.663 - 0.748i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (0.721 - 0.692i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (0.822 - 0.568i)T \)
53 \( 1 + (0.120 - 0.992i)T \)
59 \( 1 + (-0.774 + 0.632i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (0.903 + 0.428i)T \)
71 \( 1 + (0.960 + 0.278i)T \)
73 \( 1 + (-0.464 + 0.885i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.160 - 0.987i)T \)
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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

−23.00051131948141335668115943133, −21.871796755876583432612044611318, −21.53034334424414531722782616124, −20.941127284928389516372321321112, −20.04142248063703846616760911316, −18.66913220654239329820276602171, −17.89176851299867684381228529135, −17.443401608828104856897598050887, −16.47053490379830313362647691109, −15.67394277768302075586645280042, −14.93070846020713107540680458659, −14.02099228227181755425098854055, −12.79024928154517681935728296113, −12.34543492548824887055009074607, −10.93118430780410701740688710068, −10.6906478214817765690026945680, −9.43601615098519651413429420934, −8.90966334576691576610882422137, −7.78934634221397696098243448288, −6.29609032071844309342285815320, −5.64658794759101072985115034036, −5.01475436248775986549298590851, −3.99461441553099265419824284197, −2.53464119004084977166749711828, −1.5607789597875862910106942489, 0.44363236855122931256535572799, 1.92052301610668148561023442961, 2.50386354570143715683530748564, 4.2953186588519063062003602466, 5.19564302957002479674109189015, 6.068310372186581152063067105068, 6.9778570589639500101708017748, 7.65229188713113056161772856778, 8.75484764922260411215787605079, 10.068724992990060982926705262168, 10.77749737048769832096368716198, 11.3080488641787226491050683955, 12.61013709094710573390845712256, 13.35101786825283573603606061543, 13.84748245986908697263572418432, 14.87423124845512942332174950839, 16.02026052129234573659578360996, 17.08969388658209121206039236940, 17.44062596463667861491098170413, 18.34533132771547346527831765519, 18.797006794385594517957457516209, 20.125290942840689858342082973119, 20.81402638996000718470468883943, 21.74868504405791287104540564102, 22.53674570628871723688999540472

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