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Properties

Label 2-1368-152.75-c1-0-17
Degree $2$
Conductor $1368$
Sign $0.0784 - 0.996i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (−1.39 − 0.244i)2-s + (1.88 + 0.680i)4-s + 3.10i·5-s − 4.34i·7-s + (−2.45 − 1.40i)8-s + (0.757 − 4.32i)10-s + 2.65·11-s − 5.10·13-s + (−1.06 + 6.05i)14-s + (3.07 + 2.55i)16-s − 3.48·17-s + (1.86 + 3.93i)19-s + (−2.11 + 5.83i)20-s + (−3.70 − 0.648i)22-s + 5.31i·23-s + ⋯
L(s)  = 1  + (−0.984 − 0.172i)2-s + (0.940 + 0.340i)4-s + 1.38i·5-s − 1.64i·7-s + (−0.867 − 0.497i)8-s + (0.239 − 1.36i)10-s + 0.800·11-s − 1.41·13-s + (−0.283 + 1.61i)14-s + (0.768 + 0.639i)16-s − 0.846·17-s + (0.427 + 0.903i)19-s + (−0.471 + 1.30i)20-s + (−0.788 − 0.138i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.0784 - 0.996i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.0784 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7783769734\)
\(L(\frac12)\) \(\approx\) \(0.7783769734\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.244i)T \)
3 \( 1 \)
19 \( 1 + (-1.86 - 3.93i)T \)
good5 \( 1 - 3.10iT - 5T^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 + 3.48T + 17T^{2} \)
23 \( 1 - 5.31iT - 23T^{2} \)
29 \( 1 - 7.98T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 - 4.57T + 37T^{2} \)
41 \( 1 - 4.51iT - 41T^{2} \)
43 \( 1 - 6.28T + 43T^{2} \)
47 \( 1 - 7.68iT - 47T^{2} \)
53 \( 1 + 9.81T + 53T^{2} \)
59 \( 1 - 4.94iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 + 0.406iT - 67T^{2} \)
71 \( 1 + 2.40T + 71T^{2} \)
73 \( 1 - 0.373T + 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 1.95iT - 89T^{2} \)
97 \( 1 - 4.87iT - 97T^{2} \)
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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

−9.902115926609973968121041699315, −9.266050575260287965538559581636, −7.85535945265363259289847873780, −7.41857066217889828370081348888, −6.83131290738591601900150603311, −6.14088119216844664050236914449, −4.42321337377952956097299053643, −3.49065834658474789024605093691, −2.59335528368655774999752140773, −1.23314760460936406753226818248, 0.46958034222288251726594447264, 1.95818291178911896976539143633, 2.73717293273415761150502280866, 4.64550180565811628025911495129, 5.21198590903105605946176509190, 6.20929548391036506933521548056, 6.99728830732337225296071280293, 8.185763719487528858502278982429, 8.697323451575156036138592976388, 9.300520890623209049055114889049

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