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Properties

Label 2-1352-104.75-c0-0-1
Degree $2$
Conductor $1352$
Sign $-0.839 - 0.542i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
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  + (−0.866 + 0.5i)2-s + (0.900 + 1.56i)3-s + (0.499 − 0.866i)4-s + (−1.56 − 0.900i)6-s + 0.999i·8-s + (−1.12 + 1.94i)9-s + (−0.385 + 0.222i)11-s + 1.80·12-s + (−0.5 − 0.866i)16-s + (−0.222 + 0.385i)17-s − 2.24i·18-s + (1.07 + 0.623i)19-s + (0.222 − 0.385i)22-s + (−1.56 + 0.900i)24-s − 25-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.900 + 1.56i)3-s + (0.499 − 0.866i)4-s + (−1.56 − 0.900i)6-s + 0.999i·8-s + (−1.12 + 1.94i)9-s + (−0.385 + 0.222i)11-s + 1.80·12-s + (−0.5 − 0.866i)16-s + (−0.222 + 0.385i)17-s − 2.24i·18-s + (1.07 + 0.623i)19-s + (0.222 − 0.385i)22-s + (−1.56 + 0.900i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.839 - 0.542i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.839 - 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9000205079\)
\(L(\frac12)\) \(\approx\) \(0.9000205079\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.80iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.24iT - T^{2} \)
89 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.876496929870414390277917476844, −9.438065006885219566271313897810, −8.573124096628255303791893276618, −8.021776568168019192623097698532, −7.20768922957076963223730518479, −5.85139690446270254717973718050, −5.16320493656672575547000508671, −4.16241093064998717349989888705, −3.13612443397906508540646117212, −2.01612179241514000984213709369, 0.911332480734806466479675601219, 2.10793522942045647810852489390, 2.82725245319831247095593941286, 3.76970353861368201974667248916, 5.53755520355160607126709525119, 6.77494465566274789244050462359, 7.16353365546566206810993829365, 8.034408741851222636829622768845, 8.472232342508256815626938439905, 9.351998850853540000310288923532

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