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Properties

Label 2-2448-68.19-c0-0-0
Degree $2$
Conductor $2448$
Sign $0.0465 - 0.998i$
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 + 1.70i)5-s + 1.41i·13-s − i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (0.707 − 0.292i)37-s + (0.292 − 0.707i)41-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (−0.707 + 1.70i)61-s + (−2.41 + 1.00i)65-s + (0.292 + 0.707i)73-s + (1.70 − 0.707i)85-s + 1.41i·89-s + (−0.707 − 1.70i)97-s + ⋯
L(s)  = 1  + (0.707 + 1.70i)5-s + 1.41i·13-s − i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (0.707 − 0.292i)37-s + (0.292 − 0.707i)41-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (−0.707 + 1.70i)61-s + (−2.41 + 1.00i)65-s + (0.292 + 0.707i)73-s + (1.70 − 0.707i)85-s + 1.41i·89-s + (−0.707 − 1.70i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $0.0465 - 0.998i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2448} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2448,\ (\ :0),\ 0.0465 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.307652966\)
\(L(\frac12)\) \(\approx\) \(1.307652966\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−9.521459482000262600670892639988, −8.669185716516377298881916053957, −7.45731888294496748424188035062, −6.96434781874073554109776476932, −6.40125741945701451919539954377, −5.57771346058493760431955655222, −4.51018055704137215229199077185, −3.45133258893759533085060405930, −2.63774307725563441118197266478, −1.81020485108707554383956443930, 0.902374834681912290179659613410, 1.91235442680749234644166954004, 3.16600916129079813808798032797, 4.38561259870922964929869643830, 4.96124705602761381212492358714, 5.86203553782364479241778180497, 6.25490272147500911839659640002, 7.888683795123313151272462205367, 8.060478811149004698591937420730, 8.973445375018758697439802915444

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