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Properties

Label 1-79-79.25-r0-0-0
Degree $1$
Conductor $79$
Sign $0.179 + 0.983i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
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  + (−0.632 + 0.774i)2-s + (−0.845 − 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (0.948 − 0.316i)6-s + (−0.0402 + 0.999i)7-s + (0.885 + 0.464i)8-s + (0.428 + 0.903i)9-s + (0.885 − 0.464i)10-s + (0.799 + 0.600i)11-s + (−0.354 + 0.935i)12-s + (0.948 + 0.316i)13-s + (−0.748 − 0.663i)14-s + (0.568 + 0.822i)15-s + (−0.919 + 0.391i)16-s + (−0.748 + 0.663i)17-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (−0.845 − 0.534i)3-s + (−0.200 − 0.979i)4-s + (−0.919 − 0.391i)5-s + (0.948 − 0.316i)6-s + (−0.0402 + 0.999i)7-s + (0.885 + 0.464i)8-s + (0.428 + 0.903i)9-s + (0.885 − 0.464i)10-s + (0.799 + 0.600i)11-s + (−0.354 + 0.935i)12-s + (0.948 + 0.316i)13-s + (−0.748 − 0.663i)14-s + (0.568 + 0.822i)15-s + (−0.919 + 0.391i)16-s + (−0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3371545236 + 0.2811159810i\)
\(L(\frac12)\) \(\approx\) \(0.3371545236 + 0.2811159810i\)
\(L(1)\) \(\approx\) \(0.5040912773 + 0.1835999487i\)
\(L(1)\) \(\approx\) \(0.5040912773 + 0.1835999487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.632 + 0.774i)T \)
3 \( 1 + (-0.845 - 0.534i)T \)
5 \( 1 + (-0.919 - 0.391i)T \)
7 \( 1 + (-0.0402 + 0.999i)T \)
11 \( 1 + (0.799 + 0.600i)T \)
13 \( 1 + (0.948 + 0.316i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.692 - 0.721i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.996 + 0.0804i)T \)
31 \( 1 + (0.987 + 0.160i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (0.120 + 0.992i)T \)
43 \( 1 + (0.799 - 0.600i)T \)
47 \( 1 + (0.278 - 0.960i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (-0.200 + 0.979i)T \)
61 \( 1 + (-0.970 - 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.948 - 0.316i)T \)
83 \( 1 + (-0.200 - 0.979i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−30.46815523849163002116359371815, −29.78728384254369502915483935087, −28.66704736153682626130275722760, −27.58568278256890245035878550041, −26.93907855682013310980980616152, −26.21799344048367491680831274827, −24.285538334630541041842479423145, −22.75047225193801279147624247959, −22.51103240278273872834361193400, −20.8760232192758692192021916875, −20.079048347144279741878376409566, −18.81292446138632879502368377273, −17.76953869037013709286206160873, −16.56699037783028853057677404866, −15.87141385691289521512336602865, −13.98985292383092536588219944498, −12.38347758435673032592563776367, −11.22175449408658773163460285722, −10.774531595096402449653502600720, −9.38065754004048139797162588528, −7.85895734500572840953855232985, −6.521725965617618059553889651058, −4.26745415464267148700003488324, −3.50228806041729098577394719733, −0.769966534571626442836893995, 1.493757115761501461261318253161, 4.494866869644587764440021665, 5.848961670996323215781540081066, 6.938625453997076730226773757028, 8.19983850659871684086594876968, 9.320616152530399671892468128637, 11.13449689365436446140515483306, 11.97337751889789156975936073383, 13.415671015313348779532928023830, 15.21426387626807817191459790962, 15.914006704812902539711748056368, 17.07722050874872025895440369415, 18.09619272709453529291874169171, 19.04049643306854502283630963157, 19.996384983767122268829080493864, 22.04669835149496584713172515580, 23.050621544602229316461612788745, 24.03376776295270702438813180179, 24.723391783196351723408853744321, 25.903343690034429032875235418159, 27.37716579862233017659937075484, 28.23322127039952228823906575591, 28.5645462882123186421355878210, 30.37102161920240073803513531299, 31.38192652345960221987696974898

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