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Properties

Label 1-1224-1224.61-r1-0-0
Degree $1$
Conductor $1224$
Sign $0.519 + 0.854i$
Analytic cond. $131.537$
Root an. cond. $131.537$
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  + (0.130 − 0.991i)5-s + (−0.130 − 0.991i)7-s + (−0.991 + 0.130i)11-s + (−0.866 − 0.5i)13-s + (0.707 + 0.707i)19-s + (0.608 + 0.793i)23-s + (−0.965 − 0.258i)25-s + (−0.793 − 0.608i)29-s + (0.991 + 0.130i)31-s − 35-s + (−0.382 − 0.923i)37-s + (−0.793 + 0.608i)41-s + (−0.965 − 0.258i)43-s + (−0.866 + 0.5i)47-s + (−0.965 + 0.258i)49-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)5-s + (−0.130 − 0.991i)7-s + (−0.991 + 0.130i)11-s + (−0.866 − 0.5i)13-s + (0.707 + 0.707i)19-s + (0.608 + 0.793i)23-s + (−0.965 − 0.258i)25-s + (−0.793 − 0.608i)29-s + (0.991 + 0.130i)31-s − 35-s + (−0.382 − 0.923i)37-s + (−0.793 + 0.608i)41-s + (−0.965 − 0.258i)43-s + (−0.866 + 0.5i)47-s + (−0.965 + 0.258i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ 0.519 + 0.854i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5409980334 + 0.3043426008i\)
\(L(\frac12)\) \(\approx\) \(0.5409980334 + 0.3043426008i\)
\(L(1)\) \(\approx\) \(0.8233111058 - 0.2241781670i\)
\(L(1)\) \(\approx\) \(0.8233111058 - 0.2241781670i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (0.130 - 0.991i)T \)
7 \( 1 + (-0.130 - 0.991i)T \)
11 \( 1 + (-0.991 + 0.130i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.608 + 0.793i)T \)
29 \( 1 + (-0.793 - 0.608i)T \)
31 \( 1 + (0.991 + 0.130i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.793 + 0.608i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (-0.130 - 0.991i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.991 - 0.130i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.793 + 0.608i)T \)
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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

−21.08614661783625952987787987787, −19.93300337920208722963806663190, −19.110701435297069583846430811526, −18.47893680558409852765881179225, −18.0465352577163739000208882074, −17.00558860360784483988309381814, −16.123320076749469827676043351200, −15.12489757252149767778963434618, −14.96456026400328028648060434229, −13.810649225899071557030776893407, −13.153630392478266183156440986988, −12.078221194055045478802635123077, −11.51808911159550747237283071809, −10.5458531774643934087084407814, −9.868273683783596849250689865362, −9.00079079925079281176150786366, −8.07628182707996729363980761632, −7.10469095849645149573378897482, −6.49470102476130703269487346821, −5.40384504016748330963401449951, −4.80079796148503923656881142557, −3.26833209457953945102090711393, −2.712766665283131541548756647417, −1.895862657647174784083170022966, −0.15041046879678964783330780502, 0.785936288987457007967087152522, 1.79778031385657938853434246329, 3.031539871776202091978927811838, 4.000314571591338116861675583783, 5.02698147541898590905993242444, 5.45322911930069767329742199210, 6.75284309422171042221295352357, 7.767266042250952718153379097767, 8.0854861574083267610786777489, 9.4763883450312171088890379407, 9.90473112488333416598790740496, 10.776023948838329463990775044726, 11.81438694211312596058013048052, 12.6264183879335419520353332291, 13.30751873409646348335040285367, 13.863138817405765105915038078845, 14.98592502352614802800425870064, 15.780240518241367305315680224834, 16.54121847689621496262497564682, 17.2056331209077110987730184939, 17.78741458137569349036986690022, 18.85614319482549697982600422861, 19.73411377753281993835701867397, 20.337408510021471252033202277412, 20.91644428845809305139815701456

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