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Properties

Label 1-2496-2496.1229-r0-0-0
Degree $1$
Conductor $2496$
Sign $0.0723 + 0.997i$
Analytic cond. $11.5913$
Root an. cond. $11.5913$
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  + (−0.382 − 0.923i)5-s + (−0.965 + 0.258i)7-s + (−0.793 − 0.608i)11-s + (−0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (0.258 − 0.965i)23-s + (−0.707 + 0.707i)25-s + (−0.991 + 0.130i)29-s − i·31-s + (0.608 + 0.793i)35-s + (0.608 − 0.793i)37-s + (−0.965 − 0.258i)41-s + (0.991 + 0.130i)43-s − 47-s + (0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)5-s + (−0.965 + 0.258i)7-s + (−0.793 − 0.608i)11-s + (−0.866 − 0.5i)17-s + (0.608 + 0.793i)19-s + (0.258 − 0.965i)23-s + (−0.707 + 0.707i)25-s + (−0.991 + 0.130i)29-s − i·31-s + (0.608 + 0.793i)35-s + (0.608 − 0.793i)37-s + (−0.965 − 0.258i)41-s + (0.991 + 0.130i)43-s − 47-s + (0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0723 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0723 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2496\)    =    \(2^{6} \cdot 3 \cdot 13\)
Sign: $0.0723 + 0.997i$
Analytic conductor: \(11.5913\)
Root analytic conductor: \(11.5913\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2496} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2496,\ (0:\ ),\ 0.0723 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2043443236 + 0.1900635412i\)
\(L(\frac12)\) \(\approx\) \(0.2043443236 + 0.1900635412i\)
\(L(1)\) \(\approx\) \(0.6800329445 - 0.1312308909i\)
\(L(1)\) \(\approx\) \(0.6800329445 - 0.1312308909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (-0.793 - 0.608i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.608 + 0.793i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (-0.991 + 0.130i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.608 - 0.793i)T \)
41 \( 1 + (-0.965 - 0.258i)T \)
43 \( 1 + (0.991 + 0.130i)T \)
47 \( 1 - T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (-0.991 - 0.130i)T \)
61 \( 1 + (0.608 + 0.793i)T \)
67 \( 1 + (-0.130 - 0.991i)T \)
71 \( 1 + (-0.965 + 0.258i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.258 - 0.965i)T \)
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

−19.37239409359684660905079699041, −18.591424098034150872528717947228, −17.91035087225721680713901244275, −17.311276565540279328625552786490, −16.2693577141221645836190374391, −15.5861065280897375106082710923, −15.2257383683315551276275989038, −14.34631785299696732678234988420, −13.246433481253159941402894575407, −13.16491964918762931933147481221, −12.0049266046978244936769465697, −11.26843468120680367290000715440, −10.58006976345036233950278990015, −9.902802018330579729931834590648, −9.23575446693491500885506599280, −8.172886504754306578301059471747, −7.29259745739588784212947319699, −6.90256557761616199269656534474, −6.07509199083645841384737271987, −5.09842441426334472278133486086, −4.160393305939603181078633430131, −3.27756633335284653350553780041, −2.750034440160774670579499474146, −1.69450066835623133389729843349, −0.11468131150969276778832339682, 0.80380528517931381410157748382, 2.11600915613772973203911547534, 2.9906131010774171530508066102, 3.84415801227080570676515633447, 4.66938741841822706083694079411, 5.58443982840516863822194360989, 6.111231904696593766579538692477, 7.237704703532048412552396969210, 7.90243002986567415431787117459, 8.80319426368014508159172580410, 9.275824533784290580213696385542, 10.12301771294491828019825446049, 11.01111432438504263815771810115, 11.774910954056016382505824278004, 12.55243668109697172438447415380, 13.13289952331776050652248433404, 13.62623011297475346260993004697, 14.75004285246443757724476960934, 15.55403006002484154153181360939, 16.21537047475276774875288672621, 16.4881925776984998324752886217, 17.38799994114909273638560367806, 18.544194009173442792793436050403, 18.72417718536250012558189910242, 19.736150806724502078705852526071

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