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Properties

Label 2-1920-20.7-c1-0-34
Degree $2$
Conductor $1920$
Sign $0.850 + 0.525i$
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 + (2 − i)5-s + (−2 + 2i)7-s + 1.00i·9-s − 4.82i·11-s + (4.41 − 4.41i)13-s + (2.12 + 0.707i)15-s + (2.41 + 2.41i)17-s − 2.82·21-s + (−4.82 − 4.82i)23-s + (3 − 4i)25-s + (−0.707 + 0.707i)27-s + 5.65i·29-s − 8.82i·31-s + (3.41 − 3.41i)33-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.894 − 0.447i)5-s + (−0.755 + 0.755i)7-s + 0.333i·9-s − 1.45i·11-s + (1.22 − 1.22i)13-s + (0.547 + 0.182i)15-s + (0.585 + 0.585i)17-s − 0.617·21-s + (−1.00 − 1.00i)23-s + (0.600 − 0.800i)25-s + (−0.136 + 0.136i)27-s + 1.05i·29-s − 1.58i·31-s + (0.594 − 0.594i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.243247678\)
\(L(\frac12)\) \(\approx\) \(2.243247678\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + (-4.41 + 4.41i)T - 13iT^{2} \)
17 \( 1 + (-2.41 - 2.41i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (4.82 + 4.82i)T + 23iT^{2} \)
29 \( 1 - 5.65iT - 29T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 + (6.07 + 6.07i)T + 37iT^{2} \)
41 \( 1 - 6.82T + 41T^{2} \)
43 \( 1 + (-3.17 - 3.17i)T + 43iT^{2} \)
47 \( 1 + (2.82 - 2.82i)T - 47iT^{2} \)
53 \( 1 + (-0.171 + 0.171i)T - 53iT^{2} \)
59 \( 1 - 8.82T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + (-4.82 + 4.82i)T - 67iT^{2} \)
71 \( 1 + 2.34iT - 71T^{2} \)
73 \( 1 + (7.48 - 7.48i)T - 73iT^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 + (-5.17 - 5.17i)T + 83iT^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (-10.6 - 10.6i)T + 97iT^{2} \)
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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

−9.048206804871633233510325596645, −8.494336545987179775272165671143, −7.961069168951368303025836288269, −6.37324679981544697867096924300, −5.83647083271117622003369209966, −5.44942128813725968819610734156, −3.96338357931677975704060342641, −3.21016238693102485309374361947, −2.32531473344134049623209543101, −0.830082995303393776179049750998, 1.37658194746318917254880932753, 2.19438948224497527073737064795, 3.39743269108983027259656642156, 4.13337577898750993258042309792, 5.35614084928178554928463384249, 6.39302160507053714615578769583, 6.88003577587026954647290035646, 7.45658006064752160132972859511, 8.572244712993930293014300455666, 9.479339230638994785242084689248

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