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Properties

Label 2-2760-345.137-c0-0-2
Degree $2$
Conductor $2760$
Sign $0.973 + 0.229i$
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  + 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 + 27-s − 29-s − 31-s + 1.00i·35-s + (−0.707 + 0.707i)37-s + ⋯
L(s)  = 1  + 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 + 27-s − 29-s − 31-s + 1.00i·35-s + (−0.707 + 0.707i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.973 + 0.229i$
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: \((2,\ 2760,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.739954664\)
\(L(\frac12)\) \(\approx\) \(1.739954664\)
\(L(1)\) 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 \( 1 - T \)
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

−8.653048285332848437428635150986, −8.312969584211696303891483794637, −7.33354416103580858758868393966, −7.17843657656515561362973620472, −6.02971156557631318241108877641, −4.78083806288620263499399615810, −4.07673682085867526693422109651, −3.30711193704422804543908101268, −2.53658278465525417255501957901, −1.17821129963383442425022917312, 1.52041477015295031960808740563, 2.19492203667401294203760853353, 3.71060429016024908511989568801, 3.98164640975534348231637770703, 5.03968045548116239167462363518, 5.84488189149506771085838082633, 7.10587194689797668813253320072, 7.56345600610953558890487736537, 8.574338659273628140066968565483, 8.919596899802337700530580672079

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