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Properties

Label 2-3344-76.75-c1-0-93
Degree $2$
Conductor $3344$
Sign $0.802 + 0.596i$
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  + 2.58·3-s + 4.01·5-s − 1.30i·7-s + 3.66·9-s − i·11-s − 4.08i·13-s + 10.3·15-s − 5.19·17-s + (−1.73 − 4.00i)19-s − 3.36i·21-s + 0.289i·23-s + 11.1·25-s + 1.70·27-s + 5.00i·29-s + 5.32·31-s + ⋯
L(s)  = 1  + 1.49·3-s + 1.79·5-s − 0.492i·7-s + 1.22·9-s − 0.301i·11-s − 1.13i·13-s + 2.67·15-s − 1.26·17-s + (−0.397 − 0.917i)19-s − 0.734i·21-s + 0.0603i·23-s + 2.22·25-s + 0.328·27-s + 0.930i·29-s + 0.956·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.802 + 0.596i$
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: \((2,\ 3344,\ (\ :1/2),\ 0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.427111640\)
\(L(\frac12)\) \(\approx\) \(4.427111640\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 4.00i)T \)
good3 \( 1 - 2.58T + 3T^{2} \)
5 \( 1 - 4.01T + 5T^{2} \)
7 \( 1 + 1.30iT - 7T^{2} \)
13 \( 1 + 4.08iT - 13T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
23 \( 1 - 0.289iT - 23T^{2} \)
29 \( 1 - 5.00iT - 29T^{2} \)
31 \( 1 - 5.32T + 31T^{2} \)
37 \( 1 - 8.08iT - 37T^{2} \)
41 \( 1 - 1.06iT - 41T^{2} \)
43 \( 1 + 6.01iT - 43T^{2} \)
47 \( 1 + 1.68iT - 47T^{2} \)
53 \( 1 + 3.85iT - 53T^{2} \)
59 \( 1 + 2.05T + 59T^{2} \)
61 \( 1 + 3.99T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 9.25T + 79T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 - 16.7iT - 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−8.579006070582578972563045280518, −8.109940226791589871814394114740, −6.93470266667445235240722680478, −6.52545070509264646649745020923, −5.46209463985748168185574598021, −4.73422535811393985883577820852, −3.58641668831546613832151989933, −2.68126223222452392906513252636, −2.25197116056575476477522607062, −1.10406177542688729857292010706, 1.68005910775560024046790813801, 2.18950552655103601234540767497, 2.72149150521933072166312937913, 3.99436221247007217159938988068, 4.73258698395254661621682783279, 5.90858442056415004567184613702, 6.36666184094204728527064914101, 7.23762562737900667381748266118, 8.203947572131422073953561603536, 8.941042584054700559909457692013

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