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Properties

Label 2-1920-80.69-c1-0-31
Degree $2$
Conductor $1920$
Sign $0.626 + 0.779i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
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  + (0.707 + 0.707i)3-s + (0.561 − 2.16i)5-s − 4.51·7-s + 1.00i·9-s + (3.44 + 3.44i)11-s + (−0.113 − 0.113i)13-s + (1.92 − 1.13i)15-s − 5.03i·17-s + (0.992 − 0.992i)19-s + (−3.19 − 3.19i)21-s + 8.00·23-s + (−4.36 − 2.43i)25-s + (−0.707 + 0.707i)27-s + (1.01 − 1.01i)29-s − 6.42·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.251 − 0.967i)5-s − 1.70·7-s + 0.333i·9-s + (1.03 + 1.03i)11-s + (−0.0315 − 0.0315i)13-s + (0.497 − 0.292i)15-s − 1.22i·17-s + (0.227 − 0.227i)19-s + (−0.696 − 0.696i)21-s + 1.66·23-s + (−0.873 − 0.486i)25-s + (−0.136 + 0.136i)27-s + (0.188 − 0.188i)29-s − 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.626 + 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665565031\)
\(L(\frac12)\) \(\approx\) \(1.665565031\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.561 + 2.16i)T \)
good7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + (-3.44 - 3.44i)T + 11iT^{2} \)
13 \( 1 + (0.113 + 0.113i)T + 13iT^{2} \)
17 \( 1 + 5.03iT - 17T^{2} \)
19 \( 1 + (-0.992 + 0.992i)T - 19iT^{2} \)
23 \( 1 - 8.00T + 23T^{2} \)
29 \( 1 + (-1.01 + 1.01i)T - 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (-1.63 + 1.63i)T - 37iT^{2} \)
41 \( 1 + 3.35iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 + (-3.27 + 3.27i)T - 53iT^{2} \)
59 \( 1 + (-5.30 - 5.30i)T + 59iT^{2} \)
61 \( 1 + (-5.87 + 5.87i)T - 61iT^{2} \)
67 \( 1 + (1.87 + 1.87i)T + 67iT^{2} \)
71 \( 1 - 0.635iT - 71T^{2} \)
73 \( 1 + 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (6.39 + 6.39i)T + 83iT^{2} \)
89 \( 1 + 0.579iT - 89T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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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

−9.089147376919202136531884642165, −8.821428062135732386420988319967, −7.20920391575584874338765185633, −6.99692094320839555751554935142, −5.79198354248916776860118614464, −4.98414687808172498505894734327, −4.08349630387955577882156848177, −3.26158916157443685885917118017, −2.18184890095181600503262440020, −0.64880751493709392778773281428, 1.18031191450683316794439503247, 2.69133894928506950853180333664, 3.31686717265605196812534023240, 3.94057880125658085609548002135, 5.73244184194745183120331135452, 6.30567853887670908691053470074, 6.79294725842842156771592812084, 7.58066994951012333953083342380, 8.725715083334972407724139575579, 9.270643400495843080125983432648

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