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Properties

Label 2-1344-48.35-c1-0-10
Degree $2$
Conductor $1344$
Sign $-0.949 - 0.312i$
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  + (0.223 + 1.71i)3-s + (2.27 + 2.27i)5-s − 7-s + (−2.90 + 0.766i)9-s + (1.53 − 1.53i)11-s + (−3.63 − 3.63i)13-s + (−3.39 + 4.40i)15-s + 1.81i·17-s + (−4.98 + 4.98i)19-s + (−0.223 − 1.71i)21-s + 7.21i·23-s + 5.30i·25-s + (−1.96 − 4.81i)27-s + (−2.77 + 2.77i)29-s + 4.14i·31-s + ⋯
L(s)  = 1  + (0.128 + 0.991i)3-s + (1.01 + 1.01i)5-s − 0.377·7-s + (−0.966 + 0.255i)9-s + (0.463 − 0.463i)11-s + (−1.00 − 1.00i)13-s + (−0.876 + 1.13i)15-s + 0.441i·17-s + (−1.14 + 1.14i)19-s + (−0.0486 − 0.374i)21-s + 1.50i·23-s + 1.06i·25-s + (−0.377 − 0.925i)27-s + (−0.514 + 0.514i)29-s + 0.744i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.949 - 0.312i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373944787\)
\(L(\frac12)\) \(\approx\) \(1.373944787\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.223 - 1.71i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
11 \( 1 + (-1.53 + 1.53i)T - 11iT^{2} \)
13 \( 1 + (3.63 + 3.63i)T + 13iT^{2} \)
17 \( 1 - 1.81iT - 17T^{2} \)
19 \( 1 + (4.98 - 4.98i)T - 19iT^{2} \)
23 \( 1 - 7.21iT - 23T^{2} \)
29 \( 1 + (2.77 - 2.77i)T - 29iT^{2} \)
31 \( 1 - 4.14iT - 31T^{2} \)
37 \( 1 + (1.77 - 1.77i)T - 37iT^{2} \)
41 \( 1 - 0.517T + 41T^{2} \)
43 \( 1 + (-4.67 - 4.67i)T + 43iT^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + (7.94 + 7.94i)T + 53iT^{2} \)
59 \( 1 + (1.94 - 1.94i)T - 59iT^{2} \)
61 \( 1 + (-2.94 - 2.94i)T + 61iT^{2} \)
67 \( 1 + (-7.85 + 7.85i)T - 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 15.7iT - 73T^{2} \)
79 \( 1 + 5.40iT - 79T^{2} \)
83 \( 1 + (-7.12 - 7.12i)T + 83iT^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.915753947963444428036533479261, −9.519609058660223690216596870258, −8.495287148264719598545334682866, −7.58560220949145776847265239552, −6.43760515563684100629163335577, −5.85140264056497675666578784389, −5.06349119850348677644928405985, −3.68819163167062781161160742927, −3.10249760930716444202312450586, −1.99279558541291930099349461239, 0.50892226904435777059217530845, 1.97186991200961079722838480684, 2.47656211374821437511889524666, 4.28363059998477907631458211152, 5.04386131128363166814381874358, 6.15282717431349831532773751240, 6.69479819311908701345667782583, 7.50669202381299444222270400793, 8.667325972788179657973417045896, 9.171164544798071416170994173428

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