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Properties

Label 2-3344-3344.1341-c0-0-1
Degree $2$
Conductor $3344$
Sign $0.904 + 0.427i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + 1.41·7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.707 + 0.707i)11-s + (−0.965 − 0.258i)13-s + (1.36 − 0.366i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.965 + 0.258i)19-s + (0.866 + 0.500i)22-s + (0.866 − 0.5i)25-s − 26-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + 1.41·7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.5i)9-s + (0.707 + 0.707i)11-s + (−0.965 − 0.258i)13-s + (1.36 − 0.366i)14-s + (0.500 − 0.866i)16-s + (−1.22 − 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.965 + 0.258i)19-s + (0.866 + 0.500i)22-s + (0.866 − 0.5i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.904 + 0.427i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.904 + 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.607622223\)
\(L(\frac12)\) \(\approx\) \(2.607622223\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
good3 \( 1 + (0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
13 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.624271989523407940040243375534, −7.920786183723910767171203559997, −7.11887691785311886610640210231, −6.51083304372153899220482993595, −5.33711490036320635297594270613, −4.86007307145126121063228595098, −4.44224456754411159230752414125, −3.07908817477517682235618067135, −2.35061533059820885274951463492, −1.42524609987332084056173184975, 1.48457138051863687170603256680, 2.53657177147118327897214807575, 3.42347096308310842296998666861, 4.37983321610235443975583531664, 5.02223082996696492875773812048, 5.68446292344524523120078927682, 6.61741356788687394240387777216, 7.11835491760797809085237283008, 8.223187363218620401484074101422, 8.524693254779473716499066551038

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