Properties

Label 2-2760-345.137-c0-0-3
Degree $2$
Conductor $2760$
Sign $-0.229 + 0.973i$
Analytic cond. $1.37741$
Root an. cond. $1.17363$
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  i·3-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s − 9-s + (1 − i)13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 + 0.707i)23-s − 1.00i·25-s + i·27-s + 29-s − 31-s − 1.00i·35-s + (−0.707 + 0.707i)37-s + ⋯
L(s)  = 1  i·3-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s − 9-s + (1 − i)13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 + 0.707i)23-s − 1.00i·25-s + i·27-s + 29-s − 31-s − 1.00i·35-s + (−0.707 + 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(1.37741\)
Root analytic conductor: \(1.17363\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2760} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2760,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.523410713\)
\(L(\frac12)\) \(\approx\) \(1.523410713\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.596725191255395326636682940512, −7.933664515755779683306195687635, −7.53583225630553228355684786440, −6.30781155296319202107101683037, −5.84326795985066445472506305497, −5.09344736840694322924578550474, −4.00192230775584855483218061956, −2.94735370267689146060279006037, −1.62900587843445184857379240011, −1.11812944141921607095980171627, 1.78249951729130500300361076187, 2.69509354155937443838020311932, 3.62808881470386944022456487043, 4.53520158093747941873648615536, 5.42097680084432441260985158352, 5.97057968165574326904486221539, 6.78394793649546501817804873634, 7.83435920918442606124861739176, 8.776073773033709710983321649086, 9.121991183442746120155251986773

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