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Properties

Label 2-1920-80.69-c1-0-18
Degree $2$
Conductor $1920$
Sign $-0.228 - 0.973i$
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 + (−0.860 + 2.06i)5-s + 0.707·7-s + 1.00i·9-s + (1.79 + 1.79i)11-s + (3.86 + 3.86i)13-s + (−2.06 + 0.850i)15-s − 0.244i·17-s + (1.53 − 1.53i)19-s + (0.500 + 0.500i)21-s + 6.92·23-s + (−3.51 − 3.55i)25-s + (−0.707 + 0.707i)27-s + (4.89 − 4.89i)29-s − 7.60·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.384 + 0.922i)5-s + 0.267·7-s + 0.333i·9-s + (0.542 + 0.542i)11-s + (1.07 + 1.07i)13-s + (−0.533 + 0.219i)15-s − 0.0593i·17-s + (0.352 − 0.352i)19-s + (0.109 + 0.109i)21-s + 1.44·23-s + (−0.703 − 0.710i)25-s + (−0.136 + 0.136i)27-s + (0.909 − 0.909i)29-s − 1.36·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.228 - 0.973i$
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: \((2,\ 1920,\ (\ :1/2),\ -0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027567858\)
\(L(\frac12)\) \(\approx\) \(2.027567858\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.860 - 2.06i)T \)
good7 \( 1 - 0.707T + 7T^{2} \)
11 \( 1 + (-1.79 - 1.79i)T + 11iT^{2} \)
13 \( 1 + (-3.86 - 3.86i)T + 13iT^{2} \)
17 \( 1 + 0.244iT - 17T^{2} \)
19 \( 1 + (-1.53 + 1.53i)T - 19iT^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + (-4.89 + 4.89i)T - 29iT^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 + (8.47 - 8.47i)T - 37iT^{2} \)
41 \( 1 + 2.12iT - 41T^{2} \)
43 \( 1 + (-0.684 + 0.684i)T - 43iT^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
53 \( 1 + (-1.47 + 1.47i)T - 53iT^{2} \)
59 \( 1 + (-5.86 - 5.86i)T + 59iT^{2} \)
61 \( 1 + (0.0537 - 0.0537i)T - 61iT^{2} \)
67 \( 1 + (-7.85 - 7.85i)T + 67iT^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + 9.69T + 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + (-6.80 - 6.80i)T + 83iT^{2} \)
89 \( 1 - 3.07iT - 89T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−9.332910195506612455499027535350, −8.746836242951017709057616710954, −7.897276768357719344939853846508, −6.90814095004150839327569194465, −6.60147588860291885147293851690, −5.29443100827509600943728397379, −4.31241746456828694379983765790, −3.63949168786324357421280304259, −2.70865732416701807487779056905, −1.50979384736841018544526438845, 0.77254748595619994972733581871, 1.63774459138033605896946865049, 3.24905211422917802846264304496, 3.75769142501636033420884905738, 5.05531226733350548984675466350, 5.63585634214878447515511334439, 6.71500500209837007918872068541, 7.55162980359097904212613116297, 8.378081719713642061189211822843, 8.757466151740398156683129719731

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