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Properties

Label 2-1368-152.75-c1-0-49
Degree $2$
Conductor $1368$
Sign $0.458 + 0.888i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + 3.46i·7-s − 2.82·8-s − 3.16·11-s + 4.47·13-s + (4.24 + 2.44i)14-s + (−2.00 + 3.46i)16-s + 6.32·17-s + (−2 − 3.87i)19-s + (−2.23 + 3.87i)22-s − 5.47i·23-s + 5·25-s + (3.16 − 5.47i)26-s + (5.99 − 3.46i)28-s + 1.41·29-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.30i·7-s − 0.999·8-s − 0.953·11-s + 1.24·13-s + (1.13 + 0.654i)14-s + (−0.500 + 0.866i)16-s + 1.53·17-s + (−0.458 − 0.888i)19-s + (−0.476 + 0.825i)22-s − 1.14i·23-s + 25-s + (0.620 − 1.07i)26-s + (1.13 − 0.654i)28-s + 0.262·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.458 + 0.888i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.067605760\)
\(L(\frac12)\) \(\approx\) \(2.067605760\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
19 \( 1 + (2 + 3.87i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 3.16T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 6.32T + 17T^{2} \)
23 \( 1 + 5.47iT - 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 2.44iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 5.47iT - 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 4.89iT - 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 - 7.74iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 17.1iT - 89T^{2} \)
97 \( 1 - 15.4iT - 97T^{2} \)
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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

−9.617493573171367216276893190496, −8.556028679909432976816231542223, −8.341358748628071189374412406957, −6.70153371537200168976792443577, −5.85777757090426321207094128649, −5.23660582006148454904944230039, −4.30779473785544169322062344364, −2.98443064598103847496595649296, −2.52457442625581565413530075563, −0.995516195806499926593447299045, 1.04223324214299789113984332669, 3.07611112779437945429674353468, 3.79465213999867927558712885276, 4.70017757012620976440783590619, 5.67971966665726130538963715559, 6.37237442368263168952373889378, 7.40035032251638681397550806086, 7.896874418415092363863682825181, 8.566772974533234653391078738230, 9.794283396469660378325698444402

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