Properties

Label 2-1368-152.75-c1-0-27
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

\(L(1)\) \(\approx\) \(2.716176310\)
\(L(\frac12)\) \(\approx\) \(2.716176310\)
\(L(\frac{3}{2})\) 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

−9.818668496616684075195897777802, −9.338675788667482382058669967921, −7.67010643748554572706352462992, −7.41755702923505234247707008015, −6.29440667114757734894847635673, −5.53340464742357374800194075073, −5.00644075257023237888962878798, −3.47465061048594731683646401037, −2.69713690610255520972196506122, −2.20832032440615316030477072870, 0.75920423498462677745447810702, 2.15427975297060303107735857855, 3.60943318073915340568670933166, 4.32999736159725461129921851217, 5.10854856209007047353700678309, 5.70022287346697032186581625814, 7.11676906281100313548432956683, 7.57273621784529673259217677802, 8.233139404236899217410593332134, 9.531118650490601630638719071559

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