Properties

Label 2-2736-76.55-c0-0-1
Degree $2$
Conductor $2736$
Sign $0.755 + 0.654i$
Analytic cond. $1.36544$
Root an. cond. $1.16852$
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  + (1.11 − 0.642i)7-s + (1.43 − 1.20i)13-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)25-s + (−1.70 + 0.984i)31-s + 0.347·37-s + (−0.673 − 1.85i)43-s + (0.326 − 0.565i)49-s + (1.76 + 0.642i)61-s + (1.26 − 0.223i)67-s + (1.43 + 1.20i)73-s + (−0.826 + 0.984i)79-s + (0.826 − 2.27i)91-s + (0.173 − 0.984i)97-s + (−0.592 − 0.342i)103-s + ⋯
L(s)  = 1  + (1.11 − 0.642i)7-s + (1.43 − 1.20i)13-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)25-s + (−1.70 + 0.984i)31-s + 0.347·37-s + (−0.673 − 1.85i)43-s + (0.326 − 0.565i)49-s + (1.76 + 0.642i)61-s + (1.26 − 0.223i)67-s + (1.43 + 1.20i)73-s + (−0.826 + 0.984i)79-s + (0.826 − 2.27i)91-s + (0.173 − 0.984i)97-s + (−0.592 − 0.342i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(1.36544\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :0),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.419979300\)
\(L(\frac12)\) \(\approx\) \(1.419979300\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.347T + T^{2} \)
41 \( 1 + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
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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

−8.603699409765718913139388191702, −8.347864388087448544125180327458, −7.41054870532788427080806736772, −6.79916457076492296265893617331, −5.62120786449923542282946289920, −5.19212173402249348812964473101, −4.03944920814586975055848328400, −3.45609794551037068429060020729, −2.08728553481229556377015702274, −1.02642732426606904064979820242, 1.56404227040650274525021717039, 2.18229922937424077199966067396, 3.67540167275001715205225433856, 4.27442059749124439939269772986, 5.27191644468582669942185652777, 6.04564599254295062347358246441, 6.66286990599406341409289229343, 7.902445671647875336968473398533, 8.221307077590422262833838431612, 9.040124741503497829923061347772

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