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Properties

Label 2-1520-380.379-c1-0-21
Degree $2$
Conductor $1520$
Sign $0.242 - 0.970i$
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  + 1.80i·3-s + (1.73 − 1.41i)5-s − 2.12·7-s − 0.267·9-s + 4.11i·11-s + 2.72·13-s + (2.55 + 3.13i)15-s − 1.79i·17-s + (3.49 − 2.60i)19-s − 3.85i·21-s − 3.68·23-s + (0.999 − 4.89i)25-s + 4.93i·27-s + 10.5i·29-s + 4.42·31-s + ⋯
L(s)  = 1  + 1.04i·3-s + (0.774 − 0.632i)5-s − 0.804·7-s − 0.0893·9-s + 1.24i·11-s + 0.755·13-s + (0.660 + 0.808i)15-s − 0.434i·17-s + (0.801 − 0.598i)19-s − 0.840i·21-s − 0.769·23-s + (0.199 − 0.979i)25-s + 0.950i·27-s + 1.95i·29-s + 0.795·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875164995\)
\(L(\frac12)\) \(\approx\) \(1.875164995\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.41i)T \)
19 \( 1 + (-3.49 + 2.60i)T \)
good3 \( 1 - 1.80iT - 3T^{2} \)
7 \( 1 + 2.12T + 7T^{2} \)
11 \( 1 - 4.11iT - 11T^{2} \)
13 \( 1 - 2.72T + 13T^{2} \)
17 \( 1 + 1.79iT - 17T^{2} \)
23 \( 1 + 3.68T + 23T^{2} \)
29 \( 1 - 10.5iT - 29T^{2} \)
31 \( 1 - 4.42T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 3.85iT - 41T^{2} \)
43 \( 1 + 7.94T + 43T^{2} \)
47 \( 1 - 6.38T + 47T^{2} \)
53 \( 1 + 4.71T + 53T^{2} \)
59 \( 1 + 4.42T + 59T^{2} \)
61 \( 1 - 8.19T + 61T^{2} \)
67 \( 1 - 3.13iT - 67T^{2} \)
71 \( 1 - 16.5T + 71T^{2} \)
73 \( 1 - 13.3iT - 73T^{2} \)
79 \( 1 + 6.29T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.621211212215332945131746946693, −9.249069513654936502406165215824, −8.257066576262541920375186481915, −7.07504022003570817712334282578, −6.34929093809138011794768574187, −5.26009016554361286907481250273, −4.71302608104864912753465018821, −3.78798911379585341694070785094, −2.70904642172997030440259019366, −1.30864976419199691989807100399, 0.833636028936640232739233535069, 2.04904241998673656131623496445, 3.05565104451144876249603512317, 3.95690724966183086857343020627, 5.65399479862715130857305788482, 6.26683970505418616610138396683, 6.54228947948169840970286908926, 7.78491360482779670380498300096, 8.222539408207384931714442237999, 9.475720866843669962770407687992

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