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Properties

Label 2-924-21.20-c1-0-23
Degree $2$
Conductor $924$
Sign $0.654 + 0.755i$
Analytic cond. $7.37817$
Root an. cond. $2.71628$
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  + 1.73·3-s − 1.73·5-s + (2 − 1.73i)7-s + 2.99·9-s − i·11-s − 5.19i·13-s − 2.99·15-s + 1.73i·19-s + (3.46 − 2.99i)21-s − 6i·23-s − 2.00·25-s + 5.19·27-s + 9i·29-s − 3.46i·31-s − 1.73i·33-s + ⋯
L(s)  = 1  + 1.00·3-s − 0.774·5-s + (0.755 − 0.654i)7-s + 0.999·9-s − 0.301i·11-s − 1.44i·13-s − 0.774·15-s + 0.397i·19-s + (0.755 − 0.654i)21-s − 1.25i·23-s − 0.400·25-s + 1.00·27-s + 1.67i·29-s − 0.622i·31-s − 0.301i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(924\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(7.37817\)
Root analytic conductor: \(2.71628\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{924} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 924,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91481 - 0.874785i\)
\(L(\frac12)\) \(\approx\) \(1.91481 - 0.874785i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
7 \( 1 + (-2 + 1.73i)T \)
11 \( 1 + iT \)
good5 \( 1 + 1.73T + 5T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 8.66iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−10.07656944216168449215548605324, −8.884902018608363511758220283583, −8.118239792575951093321210441714, −7.75149155864955378651775263387, −6.86436422398206041886817471612, −5.46109175287522224955503737677, −4.34700557052699451998718203702, −3.63004634186886233667922654067, −2.55775686188934470944008093718, −0.976419698040574150863647470160, 1.66708033361650054981379488567, 2.66142579194151408955497801770, 4.02402424101348366872639697135, 4.51110622078658723638201313613, 5.87050646086395433828070532515, 7.16790430783297918888731522718, 7.68240907521243273707105426191, 8.559162039066090908781738931163, 9.199037023102744143216440081137, 9.931888685906614325762686253754

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