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Properties

Label 2-1344-48.35-c1-0-1
Degree $2$
Conductor $1344$
Sign $0.0101 - 0.999i$
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  + (0.0806 − 1.73i)3-s + (−1.66 − 1.66i)5-s + 7-s + (−2.98 − 0.279i)9-s + (−3.32 + 3.32i)11-s + (−0.938 − 0.938i)13-s + (−3.00 + 2.74i)15-s − 0.811i·17-s + (−0.974 + 0.974i)19-s + (0.0806 − 1.73i)21-s + 7.19i·23-s + 0.524i·25-s + (−0.723 + 5.14i)27-s + (2.21 − 2.21i)29-s + 6.74i·31-s + ⋯
L(s)  = 1  + (0.0465 − 0.998i)3-s + (−0.743 − 0.743i)5-s + 0.377·7-s + (−0.995 − 0.0930i)9-s + (−1.00 + 1.00i)11-s + (−0.260 − 0.260i)13-s + (−0.777 + 0.707i)15-s − 0.196i·17-s + (−0.223 + 0.223i)19-s + (0.0176 − 0.377i)21-s + 1.49i·23-s + 0.104i·25-s + (−0.139 + 0.990i)27-s + (0.411 − 0.411i)29-s + 1.21i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0101 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0101 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.0101 - 0.999i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.0101 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2693269476\)
\(L(\frac12)\) \(\approx\) \(0.2693269476\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0806 + 1.73i)T \)
7 \( 1 - T \)
good5 \( 1 + (1.66 + 1.66i)T + 5iT^{2} \)
11 \( 1 + (3.32 - 3.32i)T - 11iT^{2} \)
13 \( 1 + (0.938 + 0.938i)T + 13iT^{2} \)
17 \( 1 + 0.811iT - 17T^{2} \)
19 \( 1 + (0.974 - 0.974i)T - 19iT^{2} \)
23 \( 1 - 7.19iT - 23T^{2} \)
29 \( 1 + (-2.21 + 2.21i)T - 29iT^{2} \)
31 \( 1 - 6.74iT - 31T^{2} \)
37 \( 1 + (3.62 - 3.62i)T - 37iT^{2} \)
41 \( 1 + 6.63T + 41T^{2} \)
43 \( 1 + (8.58 + 8.58i)T + 43iT^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (5.01 + 5.01i)T + 53iT^{2} \)
59 \( 1 + (-3.36 + 3.36i)T - 59iT^{2} \)
61 \( 1 + (-9.07 - 9.07i)T + 61iT^{2} \)
67 \( 1 + (2.30 - 2.30i)T - 67iT^{2} \)
71 \( 1 - 1.63iT - 71T^{2} \)
73 \( 1 - 1.89iT - 73T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (5.35 + 5.35i)T + 83iT^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−9.811041480993668574990933216451, −8.602047994282281311044045357321, −8.215340224862955180732163638935, −7.39958803888793554335318143554, −6.86132226122738722130512858575, −5.42212621715931253272418157082, −4.99470456453700339812327515209, −3.73388235292125121758468520138, −2.48389991627714292470518819843, −1.38344287072033889701626130920, 0.11015676351239266297289402040, 2.50492375984597522580519259932, 3.28488062500063353875437638424, 4.22815041580939129157982700180, 5.05890524153651563454358017061, 5.99666598759380281599516450013, 6.98776582239850762106647980207, 8.073623122332726833789639031094, 8.447487867517710937424229504766, 9.459699233131768138502958061506

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