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Properties

Label 2-2368-148.115-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.0933 - 0.995i$
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  + (1.26 + 1.50i)5-s + (−0.939 + 0.342i)9-s + (−0.592 + 1.62i)13-s + (−0.439 − 1.20i)17-s + (−0.500 + 2.83i)25-s + (0.592 + 0.342i)29-s + (−0.939 − 0.342i)37-s + (1.43 + 0.524i)41-s + (−1.70 − 0.984i)45-s + (0.173 − 0.984i)49-s + (−0.766 − 0.642i)53-s + (0.233 − 0.642i)61-s + (−3.20 + 1.16i)65-s + 73-s + (0.766 − 0.642i)81-s + ⋯
L(s)  = 1  + (1.26 + 1.50i)5-s + (−0.939 + 0.342i)9-s + (−0.592 + 1.62i)13-s + (−0.439 − 1.20i)17-s + (−0.500 + 2.83i)25-s + (0.592 + 0.342i)29-s + (−0.939 − 0.342i)37-s + (1.43 + 0.524i)41-s + (−1.70 − 0.984i)45-s + (0.173 − 0.984i)49-s + (−0.766 − 0.642i)53-s + (0.233 − 0.642i)61-s + (−3.20 + 1.16i)65-s + 73-s + (0.766 − 0.642i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.0933 - 0.995i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.0933 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.258983685\)
\(L(\frac12)\) \(\approx\) \(1.258983685\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
good3 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \)
19 \( 1 + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T^{2} \)
89 \( 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.423972621127829300292350598659, −8.858755574719639842003386693475, −7.61612053666545571462432531064, −6.83335761321868984372594167820, −6.47302429080154385602623551338, −5.52835481977908697741475334364, −4.76678298161704356342603052167, −3.40664366082566603106124828864, −2.51386844624477628500330603476, −2.01071805312405424121980596000, 0.820614159735759937801580623529, 2.05370296679514957368399175505, 3.00939075648989039309178361731, 4.33400005262107273059432366472, 5.15933530484612253900517294658, 5.83418206851483286337697619661, 6.21182333926987696097949411410, 7.64934828729651163422292351993, 8.483497763360179295632335381440, 8.801584753433712126453567783338

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