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Properties

Label 2-1320-8.5-c1-0-69
Degree $2$
Conductor $1320$
Sign $-0.707 + 0.707i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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 − i)2-s + i·3-s − 2i·4-s + i·5-s + (1 + i)6-s − 2·7-s + (−2 − 2i)8-s − 9-s + (1 + i)10-s − i·11-s + 2·12-s − 4i·13-s + (−2 + 2i)14-s − 15-s − 4·16-s + 2·17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + 0.577i·3-s − i·4-s + 0.447i·5-s + (0.408 + 0.408i)6-s − 0.755·7-s + (−0.707 − 0.707i)8-s − 0.333·9-s + (0.316 + 0.316i)10-s − 0.301i·11-s + 0.577·12-s − 1.10i·13-s + (−0.534 + 0.534i)14-s − 0.258·15-s − 16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.488390461\)
\(L(\frac12)\) \(\approx\) \(1.488390461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 - iT \)
5 \( 1 - iT \)
11 \( 1 + iT \)
good7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.638471008516639167629170350671, −8.842093072953548982634440196592, −7.63258669140183318051372938942, −6.57728991501454290809645543432, −5.77418716619262750038753553825, −5.08074714770612253828345495879, −3.87027941896783486989026666301, −3.25006794480691946776905133526, −2.37433466960853583352494248748, −0.46624707050868904609824847920, 1.71109911464624447268624682019, 3.07390542123489520181533913969, 3.99047038824296281586775739118, 4.99004399972470679869031049815, 5.91126596819999716178306450491, 6.59872974742946963904696170901, 7.33815811557229557774346678912, 8.096153394494681126626379979479, 9.017785275926625322074069984505, 9.618073485672695632543889140831

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