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Properties

Label 2-1344-28.27-c1-0-7
Degree $2$
Conductor $1344$
Sign $-0.590 - 0.807i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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  − 3-s + 3.33i·5-s + (1.56 + 2.13i)7-s + 9-s − 0.936i·11-s + 1.87i·13-s − 3.33i·15-s + 5.20i·17-s + 7.12·19-s + (−1.56 − 2.13i)21-s − 0.936i·23-s − 6.12·25-s − 27-s + 2·29-s + 0.936i·33-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.49i·5-s + (0.590 + 0.807i)7-s + 0.333·9-s − 0.282i·11-s + 0.519i·13-s − 0.861i·15-s + 1.26i·17-s + 1.63·19-s + (−0.340 − 0.466i)21-s − 0.195i·23-s − 1.22·25-s − 0.192·27-s + 0.371·29-s + 0.163i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.590 - 0.807i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339414164\)
\(L(\frac12)\) \(\approx\) \(1.339414164\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good5 \( 1 - 3.33iT - 5T^{2} \)
11 \( 1 + 0.936iT - 11T^{2} \)
13 \( 1 - 1.87iT - 13T^{2} \)
17 \( 1 - 5.20iT - 17T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 + 0.936iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 + 9.06iT - 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 4.79iT - 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 + 3.86iT - 71T^{2} \)
73 \( 1 + 6.67iT - 73T^{2} \)
79 \( 1 + 2.39iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 1.46iT - 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−10.11369794403117692196035203398, −9.153423385441413610885078119427, −8.170651919418833045734745493954, −7.35475742405766762712240911349, −6.52150724173690645471708573662, −5.87972296949063765393782322068, −5.00341239518922836017698192603, −3.75367748425564844978026577241, −2.81058993810904649015915106080, −1.65091388823882080071309704809, 0.64854587076054971189441186493, 1.48349689592514403211363225787, 3.27186977650580084886127669876, 4.63884101097916424146149123477, 4.90390205939182227464557877729, 5.74028782784380603773252728103, 7.02403455940168139442476216979, 7.72206259570269768405052442017, 8.431768940799101881743593974664, 9.581163708963897463510215831939

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