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Properties

Label 2-264-24.11-c1-0-15
Degree $2$
Conductor $264$
Sign $0.993 + 0.116i$
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  + (−1.26 − 0.627i)2-s + (1.64 + 0.527i)3-s + (1.21 + 1.58i)4-s + 1.66·5-s + (−1.76 − 1.70i)6-s − 0.635i·7-s + (−0.540 − 2.77i)8-s + (2.44 + 1.74i)9-s + (−2.11 − 1.04i)10-s − i·11-s + (1.16 + 3.26i)12-s + 0.0696i·13-s + (−0.398 + 0.805i)14-s + (2.75 + 0.880i)15-s + (−1.05 + 3.85i)16-s + 2.97i·17-s + ⋯
L(s)  = 1  + (−0.896 − 0.443i)2-s + (0.952 + 0.304i)3-s + (0.606 + 0.794i)4-s + 0.745·5-s + (−0.718 − 0.695i)6-s − 0.240i·7-s + (−0.191 − 0.981i)8-s + (0.814 + 0.580i)9-s + (−0.668 − 0.330i)10-s − 0.301i·11-s + (0.335 + 0.941i)12-s + 0.0193i·13-s + (−0.106 + 0.215i)14-s + (0.710 + 0.227i)15-s + (−0.263 + 0.964i)16-s + 0.721i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.993 + 0.116i$
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: \((2,\ 264,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29166 - 0.0757404i\)
\(L(\frac12)\) \(\approx\) \(1.29166 - 0.0757404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.627i)T \)
3 \( 1 + (-1.64 - 0.527i)T \)
11 \( 1 + iT \)
good5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 0.635iT - 7T^{2} \)
13 \( 1 - 0.0696iT - 13T^{2} \)
17 \( 1 - 2.97iT - 17T^{2} \)
19 \( 1 + 2.49T + 19T^{2} \)
23 \( 1 - 3.91T + 23T^{2} \)
29 \( 1 - 3.31T + 29T^{2} \)
31 \( 1 + 9.67iT - 31T^{2} \)
37 \( 1 - 5.28iT - 37T^{2} \)
41 \( 1 + 5.12iT - 41T^{2} \)
43 \( 1 + 5.47T + 43T^{2} \)
47 \( 1 + 7.10T + 47T^{2} \)
53 \( 1 + 6.81T + 53T^{2} \)
59 \( 1 + 0.0187iT - 59T^{2} \)
61 \( 1 - 14.0iT - 61T^{2} \)
67 \( 1 + 9.82T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 3.15T + 73T^{2} \)
79 \( 1 + 16.2iT - 79T^{2} \)
83 \( 1 - 8.26iT - 83T^{2} \)
89 \( 1 + 9.87iT - 89T^{2} \)
97 \( 1 - 9.27T + 97T^{2} \)
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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

−11.75052780776022569854329633229, −10.57809883297672087484827164703, −10.02509724223089922823930206370, −9.098324259209928303098114492576, −8.357757805190326640532267424504, −7.37061177777269185895841537603, −6.14190110704255314059962641753, −4.27276721959041521585630947624, −2.98756120482364128431475314902, −1.75271234624084812215444099381, 1.64075150300395046642157790954, 2.87512539198241414307105027992, 4.96430167674738936650906366499, 6.35059362559477189384295791379, 7.15938820293958492533792648326, 8.239374104312707267212962244246, 9.074638058138915942722720735541, 9.748252882157527687901514767792, 10.65353043293738165852364058009, 11.96596137736529071237447918769

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