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Properties

Label 2-1872-52.31-c1-0-18
Degree $2$
Conductor $1872$
Sign $0.957 + 0.289i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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)5-s + (2 − 3i)13-s − 2i·17-s + 3i·25-s + 4·29-s + (−7 − 7i)37-s + (9 − 9i)41-s + 7i·49-s + 14·53-s + 10·61-s + (1 + 5i)65-s + (5 + 5i)73-s + (2 + 2i)85-s + (13 + 13i)89-s + (13 − 13i)97-s + ⋯
L(s)  = 1  + (−0.447 + 0.447i)5-s + (0.554 − 0.832i)13-s − 0.485i·17-s + 0.600i·25-s + 0.742·29-s + (−1.15 − 1.15i)37-s + (1.40 − 1.40i)41-s + i·49-s + 1.92·53-s + 1.28·61-s + (0.124 + 0.620i)65-s + (0.585 + 0.585i)73-s + (0.216 + 0.216i)85-s + (1.37 + 1.37i)89-s + (1.31 − 1.31i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.553651725\)
\(L(\frac12)\) \(\approx\) \(1.553651725\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-2 + 3i)T \)
good5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 11iT^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (7 + 7i)T + 37iT^{2} \)
41 \( 1 + (-9 + 9i)T - 41iT^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-13 - 13i)T + 89iT^{2} \)
97 \( 1 + (-13 + 13i)T - 97iT^{2} \)
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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

−9.085089984727315755970237894229, −8.443559099348125638355445623057, −7.49070219768878232430988026684, −7.03217634645471387812432730118, −5.92410276583179175432631810681, −5.25608411242825586970846982605, −4.07618261650601845342646550122, −3.33485695169846444418350429795, −2.32253219099474182216174591868, −0.75194921336979352191172044724, 0.982191842588274532153576588693, 2.24417931112986669965195765064, 3.52936919079648674560508386667, 4.31092810547757669399368250176, 5.09767789156791658935759494020, 6.20526518342110784171430923950, 6.80371590250706096906197427978, 7.86515411811517607008775054986, 8.497083101532657821282064117664, 9.098562901043967017202881254052

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