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Properties

Label 2-1368-152.75-c1-0-76
Degree $2$
Conductor $1368$
Sign $0.361 + 0.932i$
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  + (1.33 − 0.452i)2-s + (1.59 − 1.21i)4-s + 0.393i·5-s − 1.35i·7-s + (1.58 − 2.34i)8-s + (0.177 + 0.526i)10-s − 1.21·11-s + 3.30·13-s + (−0.612 − 1.81i)14-s + (1.05 − 3.85i)16-s + 0.00555·17-s + (2.48 − 3.58i)19-s + (0.476 + 0.625i)20-s + (−1.62 + 0.550i)22-s − 0.677i·23-s + ⋯
L(s)  = 1  + (0.947 − 0.320i)2-s + (0.795 − 0.606i)4-s + 0.175i·5-s − 0.511i·7-s + (0.559 − 0.829i)8-s + (0.0562 + 0.166i)10-s − 0.366·11-s + 0.917·13-s + (−0.163 − 0.484i)14-s + (0.264 − 0.964i)16-s + 0.00134·17-s + (0.570 − 0.821i)19-s + (0.106 + 0.139i)20-s + (−0.347 + 0.117i)22-s − 0.141i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.361 + 0.932i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.131232057\)
\(L(\frac12)\) \(\approx\) \(3.131232057\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.452i)T \)
3 \( 1 \)
19 \( 1 + (-2.48 + 3.58i)T \)
good5 \( 1 - 0.393iT - 5T^{2} \)
7 \( 1 + 1.35iT - 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
17 \( 1 - 0.00555T + 17T^{2} \)
23 \( 1 + 0.677iT - 23T^{2} \)
29 \( 1 + 5.04T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 - 4.23T + 37T^{2} \)
41 \( 1 + 1.96iT - 41T^{2} \)
43 \( 1 - 4.59T + 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 - 9.88iT - 59T^{2} \)
61 \( 1 - 4.99iT - 61T^{2} \)
67 \( 1 - 2.46iT - 67T^{2} \)
71 \( 1 - 0.424T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 2.05iT - 97T^{2} \)
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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

−9.590973308601812461989849725146, −8.701950136147718343359506931532, −7.46951525603520264918322466201, −6.96517107432126544031585447182, −5.94938575495578888083690803469, −5.21002748337740445711484663632, −4.20193460649817881876033509998, −3.41223027659919011909421302813, −2.39582543622636327175780614801, −1.01671504233056608190725886844, 1.62045922583332120668440607393, 2.88159804980566754942671668307, 3.74025255559336730861013341920, 4.75447005502182030034111910444, 5.66257052030583406929754921884, 6.15563731841639007453512635531, 7.27101457368816992870041532796, 7.969421656564806813535930727926, 8.785462378385025179323966178897, 9.681565647464887995607445561497

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