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Properties

Label 2-264-24.11-c1-0-1
Degree $2$
Conductor $264$
Sign $-0.844 - 0.535i$
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  + (−0.851 + 1.12i)2-s + (0.338 − 1.69i)3-s + (−0.548 − 1.92i)4-s − 3.30·5-s + (1.62 + 1.82i)6-s + 3.62i·7-s + (2.63 + 1.01i)8-s + (−2.77 − 1.14i)9-s + (2.81 − 3.73i)10-s + i·11-s + (−3.45 + 0.281i)12-s + 0.410i·13-s + (−4.09 − 3.09i)14-s + (−1.11 + 5.61i)15-s + (−3.39 + 2.11i)16-s + 7.89i·17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (0.195 − 0.980i)3-s + (−0.274 − 0.961i)4-s − 1.47·5-s + (0.665 + 0.746i)6-s + 1.37i·7-s + (0.932 + 0.360i)8-s + (−0.923 − 0.383i)9-s + (0.890 − 1.18i)10-s + 0.301i·11-s + (−0.996 + 0.0812i)12-s + 0.113i·13-s + (−1.09 − 0.826i)14-s + (−0.288 + 1.45i)15-s + (−0.849 + 0.527i)16-s + 1.91i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.844 - 0.535i$
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: \((2,\ 264,\ (\ :1/2),\ -0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0888008 + 0.305935i\)
\(L(\frac12)\) \(\approx\) \(0.0888008 + 0.305935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.851 - 1.12i)T \)
3 \( 1 + (-0.338 + 1.69i)T \)
11 \( 1 - iT \)
good5 \( 1 + 3.30T + 5T^{2} \)
7 \( 1 - 3.62iT - 7T^{2} \)
13 \( 1 - 0.410iT - 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 + 3.45T + 19T^{2} \)
23 \( 1 + 5.83T + 23T^{2} \)
29 \( 1 - 1.26T + 29T^{2} \)
31 \( 1 + 2.74iT - 31T^{2} \)
37 \( 1 + 2.82iT - 37T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 5.46T + 47T^{2} \)
53 \( 1 - 7.80T + 53T^{2} \)
59 \( 1 + 5.76iT - 59T^{2} \)
61 \( 1 - 7.43iT - 61T^{2} \)
67 \( 1 - 2.12T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + 2.31T + 73T^{2} \)
79 \( 1 - 3.31iT - 79T^{2} \)
83 \( 1 + 0.748iT - 83T^{2} \)
89 \( 1 - 2.22iT - 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−12.24729420719523058067279996486, −11.62287732394484315960489726533, −10.41777103961490774763551725406, −8.861448907588147719663220413764, −8.355970163987707585928272338521, −7.69002239835502812381969039330, −6.53409068868345419316968174544, −5.69828359844460869042868740437, −4.03219069397215699487136279143, −2.03481474573368129160094739565, 0.28695668706297186815944308845, 3.06907474782349155147126737542, 3.99100495482289423225209615423, 4.68530251715309391561008874359, 7.09305622686239862368113814021, 7.921536474987186069445196509175, 8.726821976596731667337177213463, 9.924356635385646312184373411018, 10.58724450275436535934704155947, 11.45224337461228597750785490598

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