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Properties

Label 2-4140-15.8-c1-0-22
Degree $2$
Conductor $4140$
Sign $0.666 - 0.745i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (2.23 + 0.0138i)5-s + (2.36 − 2.36i)7-s + 6.45i·11-s + (−1.76 − 1.76i)13-s + (4.25 + 4.25i)17-s + 2.13i·19-s + (−0.707 + 0.707i)23-s + (4.99 + 0.0619i)25-s + 4.27·29-s − 7.22·31-s + (5.31 − 5.25i)35-s + (−2.32 + 2.32i)37-s + 4.23i·41-s + (3.77 + 3.77i)43-s + (3.19 + 3.19i)47-s + ⋯
L(s)  = 1  + (0.999 + 0.00619i)5-s + (0.893 − 0.893i)7-s + 1.94i·11-s + (−0.489 − 0.489i)13-s + (1.03 + 1.03i)17-s + 0.490i·19-s + (−0.147 + 0.147i)23-s + (0.999 + 0.0123i)25-s + 0.794·29-s − 1.29·31-s + (0.899 − 0.888i)35-s + (−0.381 + 0.381i)37-s + 0.660i·41-s + (0.576 + 0.576i)43-s + (0.465 + 0.465i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.666 - 0.745i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.566676847\)
\(L(\frac12)\) \(\approx\) \(2.566676847\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2.23 - 0.0138i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-2.36 + 2.36i)T - 7iT^{2} \)
11 \( 1 - 6.45iT - 11T^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + (-4.25 - 4.25i)T + 17iT^{2} \)
19 \( 1 - 2.13iT - 19T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 + (2.32 - 2.32i)T - 37iT^{2} \)
41 \( 1 - 4.23iT - 41T^{2} \)
43 \( 1 + (-3.77 - 3.77i)T + 43iT^{2} \)
47 \( 1 + (-3.19 - 3.19i)T + 47iT^{2} \)
53 \( 1 + (5.04 - 5.04i)T - 53iT^{2} \)
59 \( 1 + 7.56T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 + (5.84 - 5.84i)T - 67iT^{2} \)
71 \( 1 + 6.64iT - 71T^{2} \)
73 \( 1 + (-7.79 - 7.79i)T + 73iT^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 + (-10.5 + 10.5i)T - 83iT^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (-8.04 + 8.04i)T - 97iT^{2} \)
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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

−8.386352017044683383154240575214, −7.61844667042051681819110403755, −7.27126218246311104225341351515, −6.28704268083877865910256072497, −5.49426298922870229797145635651, −4.73888962604144740599462418673, −4.17149505998320835746480369743, −2.96364999432160059237793884913, −1.81397824260747585899318569392, −1.36393619533602921017002647198, 0.73802290439277532601183251998, 1.91780499446708358772536363074, 2.70336090955339352565853935022, 3.53119453353918358944924853813, 4.91162421396888428933110155361, 5.37794451337558648741148013339, 5.92263548772546118085981683759, 6.75830940427205677717154407703, 7.69041950182219335116822030534, 8.442952472462975776673593555039

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