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Properties

Label 2-2368-148.7-c0-0-0
Degree $2$
Conductor $2368$
Sign $0.230 - 0.973i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  + (0.326 + 1.85i)5-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)13-s + (0.266 + 0.223i)17-s + (−2.37 + 0.866i)25-s + (0.766 − 1.32i)29-s + (−0.766 + 0.642i)37-s + (0.266 − 0.223i)41-s + (−0.939 + 1.62i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−1.17 + 0.984i)61-s + (1.43 + 1.20i)65-s − 73-s + (0.173 + 0.984i)81-s + ⋯
L(s)  = 1  + (0.326 + 1.85i)5-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)13-s + (0.266 + 0.223i)17-s + (−2.37 + 0.866i)25-s + (0.766 − 1.32i)29-s + (−0.766 + 0.642i)37-s + (0.266 − 0.223i)41-s + (−0.939 + 1.62i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−1.17 + 0.984i)61-s + (1.43 + 1.20i)65-s − 73-s + (0.173 + 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.372183380\)
\(L(\frac12)\) \(\approx\) \(1.372183380\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (0.766 - 0.642i)T \)
good3 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.610376184637730498531324847217, −8.352838968343865411373655440304, −7.69686457020683562998516611208, −6.99463033430265857865411069152, −6.29696529794299779581147296363, −5.64431855927942875272447820386, −4.42623890934876159070689726109, −3.45390456242015792455698121306, −2.72008074598495957476574154181, −1.70098742737307504434383176553, 1.05529167677772371402399729949, 1.76502223419924288385048491646, 3.42962542647403006156294719765, 4.34835487172419844330520920553, 4.91989539526497566001054491831, 5.80886487408463019063702639050, 6.59684707337842094870189334651, 7.54671731983100465010310600680, 8.498641235449251573344741830865, 8.987637977585599724801951959956

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