Properties

Label 2-3344-76.75-c1-0-0
Degree $2$
Conductor $3344$
Sign $-0.923 + 0.383i$
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  + 1.71·3-s − 3.66·5-s + 2.65i·7-s − 0.0489·9-s + i·11-s − 4.00i·13-s − 6.30·15-s + 5.00·17-s + (1.66 + 4.02i)19-s + 4.55i·21-s + 0.204i·23-s + 8.45·25-s − 5.23·27-s + 6.62i·29-s − 8.91·31-s + ⋯
L(s)  = 1  + 0.991·3-s − 1.64·5-s + 1.00i·7-s − 0.0163·9-s + 0.301i·11-s − 1.11i·13-s − 1.62·15-s + 1.21·17-s + (0.383 + 0.923i)19-s + 0.993i·21-s + 0.0425i·23-s + 1.69·25-s − 1.00·27-s + 1.23i·29-s − 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.923 + 0.383i$
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: \((2,\ 3344,\ (\ :1/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01122773885\)
\(L(\frac12)\) \(\approx\) \(0.01122773885\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 4.02i)T \)
good3 \( 1 - 1.71T + 3T^{2} \)
5 \( 1 + 3.66T + 5T^{2} \)
7 \( 1 - 2.65iT - 7T^{2} \)
13 \( 1 + 4.00iT - 13T^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
23 \( 1 - 0.204iT - 23T^{2} \)
29 \( 1 - 6.62iT - 29T^{2} \)
31 \( 1 + 8.91T + 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 + 1.16iT - 41T^{2} \)
43 \( 1 - 8.59iT - 43T^{2} \)
47 \( 1 + 7.02iT - 47T^{2} \)
53 \( 1 + 5.64iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + 9.02T + 67T^{2} \)
71 \( 1 - 3.63T + 71T^{2} \)
73 \( 1 + 7.45T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 3.09iT - 83T^{2} \)
89 \( 1 - 1.40iT - 89T^{2} \)
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

−8.846027191081382307382973769908, −8.265245620570352361205009383756, −7.58607890148679635382270039357, −7.37891237491518840554657586158, −5.81054980795375919162101591487, −5.34779955300519643860607409158, −4.14164361305003462131051174610, −3.26714284170144405794223125828, −3.06336906537750828843828197299, −1.67337981603753411962639597895, 0.00306455296322583060954012314, 1.32353888178430188110980659684, 2.83097766704246716779217198106, 3.48093389239663554455497656150, 4.10524522482105860728243253208, 4.77374271711595083045613969139, 6.06974657408093658609002190156, 7.15620794252453382931315409227, 7.57605211417141479382293674542, 8.026503531242010016267865839334

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