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Properties

Label 2-1920-80.69-c1-0-26
Degree $2$
Conductor $1920$
Sign $0.717 - 0.697i$
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 + (1.50 − 1.65i)5-s + 2.58·7-s + 1.00i·9-s + (4.39 + 4.39i)11-s + (0.417 + 0.417i)13-s + (2.23 − 0.110i)15-s + 4.40i·17-s + (−4.53 + 4.53i)19-s + (1.83 + 1.83i)21-s + 0.281·23-s + (−0.494 − 4.97i)25-s + (−0.707 + 0.707i)27-s + (−3.73 + 3.73i)29-s + 3.05·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.671 − 0.741i)5-s + 0.978·7-s + 0.333i·9-s + (1.32 + 1.32i)11-s + (0.115 + 0.115i)13-s + (0.576 − 0.0286i)15-s + 1.06i·17-s + (−1.04 + 1.04i)19-s + (0.399 + 0.399i)21-s + 0.0586·23-s + (−0.0989 − 0.995i)25-s + (−0.136 + 0.136i)27-s + (−0.693 + 0.693i)29-s + 0.548·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.717 - 0.697i$
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: \((2,\ 1920,\ (\ :1/2),\ 0.717 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.695697427\)
\(L(\frac12)\) \(\approx\) \(2.695697427\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 + 1.65i)T \)
good7 \( 1 - 2.58T + 7T^{2} \)
11 \( 1 + (-4.39 - 4.39i)T + 11iT^{2} \)
13 \( 1 + (-0.417 - 0.417i)T + 13iT^{2} \)
17 \( 1 - 4.40iT - 17T^{2} \)
19 \( 1 + (4.53 - 4.53i)T - 19iT^{2} \)
23 \( 1 - 0.281T + 23T^{2} \)
29 \( 1 + (3.73 - 3.73i)T - 29iT^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 + (5.26 - 5.26i)T - 37iT^{2} \)
41 \( 1 + 5.16iT - 41T^{2} \)
43 \( 1 + (-2.66 + 2.66i)T - 43iT^{2} \)
47 \( 1 + 7.45iT - 47T^{2} \)
53 \( 1 + (2.89 - 2.89i)T - 53iT^{2} \)
59 \( 1 + (4.60 + 4.60i)T + 59iT^{2} \)
61 \( 1 + (-0.211 + 0.211i)T - 61iT^{2} \)
67 \( 1 + (-7.17 - 7.17i)T + 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 4.53T + 79T^{2} \)
83 \( 1 + (-4.56 - 4.56i)T + 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 6.78iT - 97T^{2} \)
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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

−9.193122432846053511017564598009, −8.613545502776969545031916449448, −8.004934277803885254955962738726, −6.90631627513216016084371389544, −6.07654761123664032232946278346, −5.08027845495239601849316075759, −4.38012677601576764905365223462, −3.72321003435446718375252381774, −1.85020120621109428575174594788, −1.69814370919847249447273332961, 1.00002023228593297529059685215, 2.14240569755921785482907768477, 3.02054990340639642981646723742, 4.03395702324562646170240193664, 5.13167620887939193464670767466, 6.15960893621452698567353237276, 6.64320350150276183898587704952, 7.53864746053579681614517020548, 8.376024053967059980784523235321, 9.090438397725272326129119919800

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