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Properties

Label 2-3744-13.6-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.679 + 0.733i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 0.366i)5-s + (−0.866 − 0.5i)13-s + (1.5 − 0.866i)17-s + 0.732i·25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.133i)37-s + (−0.5 − 1.86i)41-s + (0.866 + 0.5i)49-s + 1.73·53-s + (−0.5 − 0.866i)61-s + (−0.5 + 0.133i)65-s + (−0.366 − 0.366i)73-s + (0.232 − 0.866i)85-s + (1.36 − 0.366i)89-s + (1.36 + 0.366i)97-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)5-s + (−0.866 − 0.5i)13-s + (1.5 − 0.866i)17-s + 0.732i·25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.133i)37-s + (−0.5 − 1.86i)41-s + (0.866 + 0.5i)49-s + 1.73·53-s + (−0.5 − 0.866i)61-s + (−0.5 + 0.133i)65-s + (−0.366 − 0.366i)73-s + (0.232 − 0.866i)85-s + (1.36 − 0.366i)89-s + (1.36 + 0.366i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.679 + 0.733i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.345483984\)
\(L(\frac12)\) \(\approx\) \(1.345483984\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - 1.73T + T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−8.713783778002706782430233486768, −7.62768534331397166490651881430, −7.39888387746198361222375502261, −6.31012111348739806615507664941, −5.34561533249153893946541770811, −5.14193745750741138359260308577, −3.93907879851800522541227720846, −3.04369909215480189073749192157, −2.12732017817196878923136902101, −0.851595361035763881976507773479, 1.32659830745744917726889701514, 2.40246568227012780599992510538, 3.25241960372024887518492999317, 4.19043893954116615040335475404, 5.09522212394471964634323058201, 5.82270676091732378725884269522, 6.60433633047011165764862177618, 7.27021155576149443204564479508, 8.053912457798255528640839187576, 8.743922743342083472551699794725

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