Properties

Label 2-2736-76.75-c1-0-37
Degree $2$
Conductor $2736$
Sign $i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4.35i·7-s − 4.35i·11-s − 7·17-s − 4.35i·19-s − 8.71i·23-s − 4·25-s + 4.35i·35-s − 13.0i·43-s − 4.35i·47-s − 12.0·49-s − 4.35i·55-s + 15·61-s + 11·73-s + 19.0·77-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.64i·7-s − 1.31i·11-s − 1.69·17-s − 0.999i·19-s − 1.81i·23-s − 0.800·25-s + 0.736i·35-s − 1.99i·43-s − 0.635i·47-s − 1.71·49-s − 0.587i·55-s + 1.92·61-s + 1.28·73-s + 2.16·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.158836061\)
\(L(\frac12)\) \(\approx\) \(1.158836061\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35iT \)
good5 \( 1 - T + 5T^{2} \)
7 \( 1 - 4.35iT - 7T^{2} \)
11 \( 1 + 4.35iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
23 \( 1 + 8.71iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 13.0iT - 43T^{2} \)
47 \( 1 + 4.35iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.789203814468410657458081175564, −8.242811409315443638822099337603, −6.83371142556904513206406479624, −6.32631396517094626619036615991, −5.57329956825449175714035398194, −4.91508712412840842965599357449, −3.78582158723682244956314339667, −2.50702635510614936132097004729, −2.25068134317812301064841386519, −0.36173015810441053873242161007, 1.33622192729485105473819296935, 2.15851289610000563168969067959, 3.59742293432366150954159283400, 4.24759561997224327573187415089, 4.95948616662942962246825509142, 6.08535659431749085987035216836, 6.82688651803480569959002336625, 7.48234465963839701103035147954, 8.020252288282131077617668391171, 9.253722077343463584061172115274

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