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Properties

Label 2-2340-780.467-c0-0-1
Degree $2$
Conductor $2340$
Sign $-0.662 - 0.749i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
Motivic weight $0$
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)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s + 1.00i·10-s − 13-s − 1.00·16-s + (−1.41 + 1.41i)17-s + (−0.707 + 0.707i)20-s + 1.00i·25-s + (−0.707 − 0.707i)26-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 2.00·34-s + (1 − i)37-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)8-s + 1.00i·10-s − 13-s − 1.00·16-s + (−1.41 + 1.41i)17-s + (−0.707 + 0.707i)20-s + 1.00i·25-s + (−0.707 − 0.707i)26-s + 1.41·29-s + (−0.707 − 0.707i)32-s − 2.00·34-s + (1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.662 - 0.749i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.709093111\)
\(L(\frac12)\) \(\approx\) \(1.709093111\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.265064264862928610265295698414, −8.632332746845859127006770126816, −7.66350646484455311833333029559, −7.01066306601757648089942576097, −6.26221289577841511216967252075, −5.76111682455943732734931546728, −4.68478505993463051904074843986, −4.00582112486346243382547084672, −2.79356229032005110392469651048, −2.14592049679446770969536672138, 0.907488786209727316730007950170, 2.30513384733070205137195182861, 2.77180630588933270405723835382, 4.33803645540880965960298904238, 4.73920655161142613111988552854, 5.50725697215815301442372474380, 6.42079015610059934453004134586, 7.10013902701638260583374166155, 8.338845319710918141714374656375, 9.224037748194297572363805052947

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