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Properties

Label 2-1344-48.11-c1-0-43
Degree $2$
Conductor $1344$
Sign $-0.941 + 0.335i$
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  + (1.68 − 0.399i)3-s + (−2.09 + 2.09i)5-s − 7-s + (2.68 − 1.34i)9-s + (−3.61 − 3.61i)11-s + (−2.99 + 2.99i)13-s + (−2.69 + 4.37i)15-s + 2.25i·17-s + (−2.89 − 2.89i)19-s + (−1.68 + 0.399i)21-s − 6.47i·23-s − 3.80i·25-s + (3.97 − 3.34i)27-s + (−7.29 − 7.29i)29-s + 6.92i·31-s + ⋯
L(s)  = 1  + (0.973 − 0.230i)3-s + (−0.938 + 0.938i)5-s − 0.377·7-s + (0.893 − 0.449i)9-s + (−1.08 − 1.08i)11-s + (−0.830 + 0.830i)13-s + (−0.696 + 1.12i)15-s + 0.545i·17-s + (−0.665 − 0.665i)19-s + (−0.367 + 0.0872i)21-s − 1.35i·23-s − 0.761i·25-s + (0.765 − 0.643i)27-s + (−1.35 − 1.35i)29-s + 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.941 + 0.335i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2046068426\)
\(L(\frac12)\) \(\approx\) \(0.2046068426\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.399i)T \)
7 \( 1 + T \)
good5 \( 1 + (2.09 - 2.09i)T - 5iT^{2} \)
11 \( 1 + (3.61 + 3.61i)T + 11iT^{2} \)
13 \( 1 + (2.99 - 2.99i)T - 13iT^{2} \)
17 \( 1 - 2.25iT - 17T^{2} \)
19 \( 1 + (2.89 + 2.89i)T + 19iT^{2} \)
23 \( 1 + 6.47iT - 23T^{2} \)
29 \( 1 + (7.29 + 7.29i)T + 29iT^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 + (-2.12 - 2.12i)T + 37iT^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (-0.598 + 0.598i)T - 43iT^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 + (2.98 - 2.98i)T - 53iT^{2} \)
59 \( 1 + (1.85 + 1.85i)T + 59iT^{2} \)
61 \( 1 + (-3.40 + 3.40i)T - 61iT^{2} \)
67 \( 1 + (7.71 + 7.71i)T + 67iT^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 - 9.70iT - 73T^{2} \)
79 \( 1 - 5.37iT - 79T^{2} \)
83 \( 1 + (4.46 - 4.46i)T - 83iT^{2} \)
89 \( 1 - 4.75T + 89T^{2} \)
97 \( 1 - 3.98T + 97T^{2} \)
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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

−9.090973487132211102274762800414, −8.318224870731599918467408781832, −7.72471498953860689658921710590, −6.90815088169334237262621883399, −6.28747261358102188481739161095, −4.77350312013154925759883784989, −3.79466680040777466691277138131, −3.00493638490683164114540901989, −2.24013876242464491696189148318, −0.06799844108371592137456095407, 1.84536909306836895364612750660, 3.00397802147257210275427363779, 3.94309613801530962713686834210, 4.80149357972019564740504725386, 5.49074846025877180653605787319, 7.26667635453988242962370475692, 7.58532320383233844096693421618, 8.265212057023034509086684215871, 9.177104349528278643932260915289, 9.830290554800157262793253364266

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