Properties

Label 2-1320-5.4-c1-0-6
Degree $2$
Conductor $1320$
Sign $0.139 - 0.990i$
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  − i·3-s + (−0.311 + 2.21i)5-s − 2.90i·7-s − 9-s − 11-s + 3.52i·13-s + (2.21 + 0.311i)15-s + 7.33i·17-s − 3.80·19-s − 2.90·21-s + (−4.80 − 1.37i)25-s + i·27-s + 8.42·29-s − 1.37·31-s + i·33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.139 + 0.990i)5-s − 1.09i·7-s − 0.333·9-s − 0.301·11-s + 0.977i·13-s + (0.571 + 0.0803i)15-s + 1.77i·17-s − 0.873·19-s − 0.633·21-s + (−0.961 − 0.275i)25-s + 0.192i·27-s + 1.56·29-s − 0.247·31-s + 0.174i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.065324432\)
\(L(\frac12)\) \(\approx\) \(1.065324432\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (0.311 - 2.21i)T \)
11 \( 1 + T \)
good7 \( 1 + 2.90iT - 7T^{2} \)
13 \( 1 - 3.52iT - 13T^{2} \)
17 \( 1 - 7.33iT - 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 8.42T + 29T^{2} \)
31 \( 1 + 1.37T + 31T^{2} \)
37 \( 1 - 3.05iT - 37T^{2} \)
41 \( 1 + 3.18T + 41T^{2} \)
43 \( 1 - 2.90iT - 43T^{2} \)
47 \( 1 - 5.80iT - 47T^{2} \)
53 \( 1 - 8.42iT - 53T^{2} \)
59 \( 1 - 8.42T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 1.71iT - 73T^{2} \)
79 \( 1 + 2.56T + 79T^{2} \)
83 \( 1 + 8.76iT - 83T^{2} \)
89 \( 1 - 5.86T + 89T^{2} \)
97 \( 1 + 6.75iT - 97T^{2} \)
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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

−10.24351444834498035291028606231, −8.831598838669536491377039876497, −8.057964811376086944716021601322, −7.29715209787243210759423841321, −6.56328404974382329962580139176, −6.07139868237772262074548185668, −4.48187925410180858979160473366, −3.78717736505831848870054951024, −2.60951780441236862543805027147, −1.44215256655231040176245244934, 0.44433502560409759892080408373, 2.27207150348960555401332890461, 3.23332149944379316736874847088, 4.53972362805209278950085706965, 5.19695785891404529443696917720, 5.78072387244093963357085470613, 7.02030583493400057771611653857, 8.202077609863913971996039447338, 8.614746459466346563295431919065, 9.399659828379070234440703690139

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