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Properties

Label 1-2912-2912.2645-r0-0-0
Degree $1$
Conductor $2912$
Sign $0.933 - 0.357i$
Analytic cond. $13.5232$
Root an. cond. $13.5232$
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  + (0.965 + 0.258i)3-s + (0.707 − 0.707i)5-s + (0.866 + 0.5i)9-s + (0.965 + 0.258i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 − 0.5i)23-s − i·25-s + (0.707 + 0.707i)27-s + (−0.965 − 0.258i)29-s + i·31-s + (0.866 + 0.5i)33-s + (−0.965 − 0.258i)37-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (0.707 − 0.707i)5-s + (0.866 + 0.5i)9-s + (0.965 + 0.258i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (0.866 − 0.5i)23-s − i·25-s + (0.707 + 0.707i)27-s + (−0.965 − 0.258i)29-s + i·31-s + (0.866 + 0.5i)33-s + (−0.965 − 0.258i)37-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.933 - 0.357i$
Analytic conductor: \(13.5232\)
Root analytic conductor: \(13.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2912,\ (0:\ ),\ 0.933 - 0.357i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.357708332 - 0.6211750937i\)
\(L(\frac12)\) \(\approx\) \(3.357708332 - 0.6211750937i\)
\(L(1)\) \(\approx\) \(1.880192366 - 0.1320379573i\)
\(L(1)\) \(\approx\) \(1.880192366 - 0.1320379573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.965 + 0.258i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (-0.258 + 0.965i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−19.258638509144887164984409502142, −18.510256161325149291630667486512, −17.97545708129120172471009311638, −17.06123921491752849932019440856, −16.519721317053458328257716162444, −15.25769518215541580147557428770, −14.91254279585244933156471953984, −14.25256080656845908655319491856, −13.589307562081343442008898862920, −13.09754497588708375532524018878, −12.12043107542715775428088506342, −11.37681434657984976684945572998, −10.44616549466448842338268671191, −9.73308467186331754801944535036, −9.19536475481226029904363489432, −8.44819348342885359998180902694, −7.466821645535032731425112485441, −7.00651156286479557409907269767, −6.10670617449603358962249982786, −5.446519892688177269487849563505, −4.12536303693681296357223826241, −3.42604278949308711867178502871, −2.8466117465513072970950187642, −1.74185954649743590036113574564, −1.28184420260292311664859018604, 1.02349126963985887401419992746, 1.69788206424551744558651838311, 2.66675542127568682616465800686, 3.4260956385439570751051204674, 4.362578680013758055724840502235, 5.04105514794391903716207348100, 5.80662483439692927886187855131, 7.01965545812053867788585502972, 7.41206295412471115562718324123, 8.64194765701466940555170422600, 8.99740623858930431576489100073, 9.62382574780809970634987872149, 10.23837017075336165545233746651, 11.251375477265560336356762961755, 12.22070949058876939169688466131, 12.7637783253481029783730229518, 13.6612137772709118629436251557, 14.12907496594869304936735277702, 14.65094918563171578695644710043, 15.71533498119555002275155803306, 16.13626273454639939793725429718, 17.038827369054766979416310482896, 17.55493587674951894496660471820, 18.55369174505758518987890795574, 19.10983148237380297487177489025

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