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Properties

Label 2-3344-76.75-c1-0-30
Degree $2$
Conductor $3344$
Sign $0.852 - 0.523i$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
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  + 0.924·3-s − 2.57·5-s − 0.765i·7-s − 2.14·9-s − i·11-s − 2.60i·13-s − 2.37·15-s − 3.74·17-s + (2.28 + 3.71i)19-s − 0.707i·21-s + 3.60i·23-s + 1.62·25-s − 4.75·27-s + 3.77i·29-s + 7.30·31-s + ⋯
L(s)  = 1  + 0.533·3-s − 1.15·5-s − 0.289i·7-s − 0.714·9-s − 0.301i·11-s − 0.723i·13-s − 0.614·15-s − 0.909·17-s + (0.523 + 0.852i)19-s − 0.154i·21-s + 0.751i·23-s + 0.324·25-s − 0.915·27-s + 0.700i·29-s + 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(26.7019\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272158337\)
\(L(\frac12)\) \(\approx\) \(1.272158337\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + iT \)
19 \( 1 + (-2.28 - 3.71i)T \)
good3 \( 1 - 0.924T + 3T^{2} \)
5 \( 1 + 2.57T + 5T^{2} \)
7 \( 1 + 0.765iT - 7T^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 - 3.77iT - 29T^{2} \)
31 \( 1 - 7.30T + 31T^{2} \)
37 \( 1 - 8.15iT - 37T^{2} \)
41 \( 1 + 5.25iT - 41T^{2} \)
43 \( 1 - 0.574iT - 43T^{2} \)
47 \( 1 - 1.77iT - 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 - 4.11T + 59T^{2} \)
61 \( 1 - 9.17T + 61T^{2} \)
67 \( 1 - 1.66T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 2.32T + 73T^{2} \)
79 \( 1 - 9.04T + 79T^{2} \)
83 \( 1 + 17.2iT - 83T^{2} \)
89 \( 1 - 3.39iT - 89T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−8.341179630598702797484216389818, −8.177439556642995224170287054177, −7.39767995948419611355404471212, −6.56721789177905501936418781454, −5.60912902120965940619792794830, −4.80031200700753624041821854216, −3.68821946842615736665380287370, −3.38180729506951966583170990156, −2.30006327914667958097864712347, −0.793103644596317828390153343691, 0.50718613090625759187154595971, 2.24544380246608542170592058291, 2.85020927313309137661589504330, 3.98522186038325167091428083343, 4.44247853714527303141367019970, 5.47360962498186939834728165912, 6.48707199070272762821269016637, 7.14849375880822812841188749991, 7.961540308819729157847363618784, 8.509259256928513694052839675555

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