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Properties

Label 2-264-24.11-c1-0-23
Degree $2$
Conductor $264$
Sign $0.984 + 0.173i$
Analytic cond. $2.10805$
Root an. cond. $1.45191$
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  + (−1.39 + 0.207i)2-s + (−1 + 1.41i)3-s + (1.91 − 0.579i)4-s + 3.95·5-s + (1.10 − 2.18i)6-s − 3.95i·7-s + (−2.55 + 1.20i)8-s + (−1.00 − 2.82i)9-s + (−5.53 + 0.819i)10-s − i·11-s + (−1.09 + 3.28i)12-s − 3.95i·13-s + (0.819 + 5.53i)14-s + (−3.95 + 5.59i)15-s + (3.32 − 2.21i)16-s − 0.828i·17-s + ⋯
L(s)  = 1  + (−0.989 + 0.146i)2-s + (−0.577 + 0.816i)3-s + (0.957 − 0.289i)4-s + 1.76·5-s + (0.451 − 0.892i)6-s − 1.49i·7-s + (−0.904 + 0.426i)8-s + (−0.333 − 0.942i)9-s + (−1.75 + 0.259i)10-s − 0.301i·11-s + (−0.316 + 0.948i)12-s − 1.09i·13-s + (0.219 + 1.47i)14-s + (−1.02 + 1.44i)15-s + (0.832 − 0.554i)16-s − 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889971 - 0.0778719i\)
\(L(\frac12)\) \(\approx\) \(0.889971 - 0.0778719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.207i)T \)
3 \( 1 + (1 - 1.41i)T \)
11 \( 1 + iT \)
good5 \( 1 - 3.95T + 5T^{2} \)
7 \( 1 + 3.95iT - 7T^{2} \)
13 \( 1 + 3.95iT - 13T^{2} \)
17 \( 1 + 0.828iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3.95T + 23T^{2} \)
29 \( 1 - 2.31T + 29T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 - 3.27iT - 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + 0.828T + 43T^{2} \)
47 \( 1 + 7.23T + 47T^{2} \)
53 \( 1 - 0.678T + 53T^{2} \)
59 \( 1 - 1.17iT - 59T^{2} \)
61 \( 1 - 3.95iT - 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 + 7.17T + 73T^{2} \)
79 \( 1 - 0.678iT - 79T^{2} \)
83 \( 1 - 3.65iT - 83T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + 7.31T + 97T^{2} \)
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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

−11.44614200915399502468876953974, −10.42581507024451194178908494924, −10.16703346003812157038808418307, −9.475381309268172222497469257911, −8.222341128138672993004581726901, −6.82996541027223084522924817603, −6.02179043036458532451406673812, −5.03049085108137712333375029654, −3.12070761557805386914521606110, −1.11050741529146670158614432169, 1.83133722615811746429478525597, 2.36677489650181104392849370698, 5.41867072720143580690556778889, 6.10588156966732732613859185755, 6.88220780404045633957425223169, 8.308275211791196073750057548858, 9.246095894735061371541557211085, 9.869652053943230032997173775313, 11.03104276971443353944662118020, 11.99069268367035098501987642078

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