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Properties

Label 2-2448-51.38-c0-0-1
Degree $2$
Conductor $2448$
Sign $0.739 + 0.672i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)11-s + 13-s + (0.707 − 0.707i)17-s + i·19-s + (0.707 − 0.707i)23-s + (1 − i)31-s + (−0.707 + 0.707i)41-s − i·43-s − 1.41i·47-s − i·49-s + 1.41·53-s + 1.00·55-s + 1.41·59-s + (−0.707 − 0.707i)65-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)11-s + 13-s + (0.707 − 0.707i)17-s + i·19-s + (0.707 − 0.707i)23-s + (1 − i)31-s + (−0.707 + 0.707i)41-s − i·43-s − 1.41i·47-s − i·49-s + 1.41·53-s + 1.00·55-s + 1.41·59-s + (−0.707 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.739 + 0.672i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.739 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.055933890\)
\(L(\frac12)\) \(\approx\) \(1.055933890\)
\(L(1)\) 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 \( 1 \)
17 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + (-1 - i)T + iT^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.715083963459840132604553550763, −8.406640310645486745186462279794, −7.59816961146099513485324432209, −6.86219006148540239009498691080, −5.79969045757943909434052906769, −5.05128907597883798497447012221, −4.25013307392428045202268560251, −3.44054000229098963518023922031, −2.24819716823621039536503776508, −0.858527810319889120661194996620, 1.21325685058925526138037382116, 2.87935065599888893064300213503, 3.33203562913442259413768120866, 4.33015473431315924906784881102, 5.39062401635132954183493919877, 6.13147235620038074475792650881, 7.01319821744920349831731897687, 7.66656008377968105325256616129, 8.442026040737310604461966717809, 9.012116979243225005684270245240

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