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Properties

Label 2-1344-48.35-c1-0-20
Degree $2$
Conductor $1344$
Sign $0.386 - 0.922i$
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  + (1.71 − 0.262i)3-s + (2.76 + 2.76i)5-s − 7-s + (2.86 − 0.899i)9-s + (−3.92 + 3.92i)11-s + (−1.26 − 1.26i)13-s + (5.45 + 4.00i)15-s + 7.10i·17-s + (0.652 − 0.652i)19-s + (−1.71 + 0.262i)21-s − 3.98i·23-s + 10.2i·25-s + (4.66 − 2.29i)27-s + (0.280 − 0.280i)29-s + 2.51i·31-s + ⋯
L(s)  = 1  + (0.988 − 0.151i)3-s + (1.23 + 1.23i)5-s − 0.377·7-s + (0.954 − 0.299i)9-s + (−1.18 + 1.18i)11-s + (−0.351 − 0.351i)13-s + (1.40 + 1.03i)15-s + 1.72i·17-s + (0.149 − 0.149i)19-s + (−0.373 + 0.0573i)21-s − 0.830i·23-s + 2.05i·25-s + (0.897 − 0.440i)27-s + (0.0520 − 0.0520i)29-s + 0.452i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.386 - 0.922i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.617861156\)
\(L(\frac12)\) \(\approx\) \(2.617861156\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.262i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.76 - 2.76i)T + 5iT^{2} \)
11 \( 1 + (3.92 - 3.92i)T - 11iT^{2} \)
13 \( 1 + (1.26 + 1.26i)T + 13iT^{2} \)
17 \( 1 - 7.10iT - 17T^{2} \)
19 \( 1 + (-0.652 + 0.652i)T - 19iT^{2} \)
23 \( 1 + 3.98iT - 23T^{2} \)
29 \( 1 + (-0.280 + 0.280i)T - 29iT^{2} \)
31 \( 1 - 2.51iT - 31T^{2} \)
37 \( 1 + (-2.94 + 2.94i)T - 37iT^{2} \)
41 \( 1 - 5.39T + 41T^{2} \)
43 \( 1 + (-1.15 - 1.15i)T + 43iT^{2} \)
47 \( 1 - 5.00T + 47T^{2} \)
53 \( 1 + (6.25 + 6.25i)T + 53iT^{2} \)
59 \( 1 + (-1.75 + 1.75i)T - 59iT^{2} \)
61 \( 1 + (2.02 + 2.02i)T + 61iT^{2} \)
67 \( 1 + (-0.986 + 0.986i)T - 67iT^{2} \)
71 \( 1 + 5.74iT - 71T^{2} \)
73 \( 1 + 0.913iT - 73T^{2} \)
79 \( 1 + 3.77iT - 79T^{2} \)
83 \( 1 + (9.11 + 9.11i)T + 83iT^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−9.902246206548427343164353382563, −9.110061841577012155120913588954, −8.023504370858546243561656068003, −7.34583737807280851211863499219, −6.57241819939393605548013929875, −5.82007263168154659177768443907, −4.59804030497620307316428454493, −3.34950854494119519051608883904, −2.49383719229142705824857363468, −1.91971587268186923221170833311, 0.949222498906630831724738688037, 2.33266357293691127021640353226, 3.03178036773452917268223213755, 4.43209125374141110262471914038, 5.26602352746308031771111926289, 5.90626301152736902991699088922, 7.20704471055615947940071389618, 8.023072239248528058293911894217, 8.832200150969163802329751735044, 9.459440493340607241694318352976

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