Properties

Label 2-1344-48.11-c1-0-28
Degree $2$
Conductor $1344$
Sign $0.935 - 0.352i$
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  + (0.714 + 1.57i)3-s + (1.31 − 1.31i)5-s + 7-s + (−1.97 + 2.25i)9-s + (1.23 + 1.23i)11-s + (4.57 − 4.57i)13-s + (3.02 + 1.13i)15-s − 5.82i·17-s + (2.27 + 2.27i)19-s + (0.714 + 1.57i)21-s + 2.63i·23-s + 1.51i·25-s + (−4.97 − 1.50i)27-s + (−4.30 − 4.30i)29-s − 5.81i·31-s + ⋯
L(s)  = 1  + (0.412 + 0.910i)3-s + (0.590 − 0.590i)5-s + 0.377·7-s + (−0.659 + 0.751i)9-s + (0.372 + 0.372i)11-s + (1.26 − 1.26i)13-s + (0.781 + 0.294i)15-s − 1.41i·17-s + (0.521 + 0.521i)19-s + (0.155 + 0.344i)21-s + 0.549i·23-s + 0.303i·25-s + (−0.956 − 0.290i)27-s + (−0.799 − 0.799i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.935 - 0.352i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.378653886\)
\(L(\frac12)\) \(\approx\) \(2.378653886\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.714 - 1.57i)T \)
7 \( 1 - T \)
good5 \( 1 + (-1.31 + 1.31i)T - 5iT^{2} \)
11 \( 1 + (-1.23 - 1.23i)T + 11iT^{2} \)
13 \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 + (-2.27 - 2.27i)T + 19iT^{2} \)
23 \( 1 - 2.63iT - 23T^{2} \)
29 \( 1 + (4.30 + 4.30i)T + 29iT^{2} \)
31 \( 1 + 5.81iT - 31T^{2} \)
37 \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + (2.12 - 2.12i)T - 43iT^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + (0.573 - 0.573i)T - 53iT^{2} \)
59 \( 1 + (8.42 + 8.42i)T + 59iT^{2} \)
61 \( 1 + (6.80 - 6.80i)T - 61iT^{2} \)
67 \( 1 + (1.90 + 1.90i)T + 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 - 0.516iT - 73T^{2} \)
79 \( 1 - 3.10iT - 79T^{2} \)
83 \( 1 + (2.91 - 2.91i)T - 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 - 8.09T + 97T^{2} \)
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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

−9.505529483500971301352727168915, −9.128278812641548723717189598937, −8.082181978532350289486318675549, −7.55783908741519542787811153242, −5.96210675910094265273574504867, −5.47544164388302632662680816466, −4.56193223340994384615255711834, −3.62513044151030503900418852645, −2.59688960766049518705622741033, −1.17638340628480182380190847331, 1.30431572016503203389411814410, 2.13032700548036175083650495496, 3.31754568932048796674170004434, 4.24545676611718685619875587225, 5.83268560587512213215018169546, 6.28702273834271983301879097841, 7.03145345442798325577495057489, 7.930365091958662491550647036603, 8.960477577643987172802427400908, 9.090685304137755551270286913563

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