Properties

Label 2-1344-12.11-c1-0-30
Degree $2$
Conductor $1344$
Sign $0.985 + 0.169i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 + 0.292i)3-s − 0.585i·5-s i·7-s + (2.82 + i)9-s + 2·11-s − 0.585·13-s + (0.171 − i)15-s − 0.828i·17-s + 2.24i·19-s + (0.292 − 1.70i)21-s + 4·23-s + 4.65·25-s + (4.53 + 2.53i)27-s + 0.828i·29-s − 6.82i·31-s + ⋯
L(s)  = 1  + (0.985 + 0.169i)3-s − 0.261i·5-s − 0.377i·7-s + (0.942 + 0.333i)9-s + 0.603·11-s − 0.162·13-s + (0.0442 − 0.258i)15-s − 0.200i·17-s + 0.514i·19-s + (0.0639 − 0.372i)21-s + 0.834·23-s + 0.931·25-s + (0.872 + 0.487i)27-s + 0.153i·29-s − 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.556870607\)
\(L(\frac12)\) \(\approx\) \(2.556870607\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.70 - 0.292i)T \)
7 \( 1 + iT \)
good5 \( 1 + 0.585iT - 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 + 0.828iT - 17T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 6.48iT - 43T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 - 9.31iT - 53T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 8.82T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 8.58T + 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.238918933191393368884044222865, −9.051594087955044777663994040781, −7.965686413598748645127029478433, −7.32596818286730315528240634969, −6.47274190361606185161668938533, −5.22098639693271642780626291486, −4.29799595698970107358947150230, −3.52217112842852372767720916966, −2.43736086931292882411768347360, −1.17377222708132401030632508982, 1.31887958695837183125099705606, 2.56235887463147169955024995061, 3.33698125326580778180454763063, 4.38620411280687931217697973946, 5.39352198867615987436534072099, 6.75970067428874770537691831875, 7.00932354781171283490275068874, 8.214184326852828690772416588645, 8.802662347771955045949803773413, 9.434745565021708967371343304200

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