Properties

Label 2-1520-5.4-c1-0-34
Degree $2$
Conductor $1520$
Sign $0.447 + 0.894i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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 − 2i)5-s − 2i·7-s + 3·9-s + 4·11-s + 2i·13-s + 4i·17-s + 19-s − 6i·23-s + (−3 + 4i)25-s + 6·29-s + 4·31-s + (−4 + 2i)35-s − 10i·37-s − 10·41-s + 2i·43-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s − 0.755i·7-s + 9-s + 1.20·11-s + 0.554i·13-s + 0.970i·17-s + 0.229·19-s − 1.25i·23-s + (−0.600 + 0.800i)25-s + 1.11·29-s + 0.718·31-s + (−0.676 + 0.338i)35-s − 1.64i·37-s − 1.56·41-s + 0.304i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.791687987\)
\(L(\frac12)\) \(\approx\) \(1.791687987\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1 + 2i)T \)
19 \( 1 - T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 18iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−9.252687208463408578968196274943, −8.570691793726506884393445217455, −7.76788643969557361212284973950, −6.85927486385040849304503143387, −6.28470601196291565371436022421, −4.84547176461084993520862042114, −4.24941557729357117797711864981, −3.64562891955454834156087366130, −1.79689909352392986389703549877, −0.853342878682414649281281933757, 1.28701161864211654620757389450, 2.69093856588231215258699587545, 3.53009422507654082277128363168, 4.50484151717264425960352891776, 5.52323719735619434916334743381, 6.62479186634131935246910711123, 7.01972002348104482095709611608, 7.967209249590351443092100967014, 8.806023744950521977225630123010, 9.790974460603350459577491348606

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