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Properties

Label 2-3240-9.4-c1-0-21
Degree $2$
Conductor $3240$
Sign $0.766 - 0.642i$
Analytic cond. $25.8715$
Root an. cond. $5.08640$
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  + (0.5 − 0.866i)5-s + (2 + 3.46i)7-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s − 2·17-s + 4·19-s + (2 − 3.46i)23-s + (−0.499 − 0.866i)25-s + (−1 − 1.73i)29-s + (4 − 6.92i)31-s + 3.99·35-s + 6·37-s + (−3 + 5.19i)41-s + (4 + 6.92i)43-s + (2 + 3.46i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s − 0.485·17-s + 0.917·19-s + (0.417 − 0.722i)23-s + (−0.0999 − 0.173i)25-s + (−0.185 − 0.321i)29-s + (0.718 − 1.24i)31-s + 0.676·35-s + 0.986·37-s + (−0.468 + 0.811i)41-s + (0.609 + 1.05i)43-s + (0.291 + 0.505i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(25.8715\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331231637\)
\(L(\frac12)\) \(\approx\) \(2.331231637\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.775738221971129928713017847475, −8.055497835935780895990818239749, −7.39564614325142566789176250416, −6.25473549548713818342062133531, −5.79283828259738763106229961254, −4.77255057291672011978712871596, −4.39401867612762282632330780686, −2.92894620361717558564372310535, −2.16028693128308461113817397367, −1.15445220581255516115920080329, 0.855327904736472794737847602444, 1.71580801263595136428293652893, 3.14499468958761546637218678166, 3.79866559309019276402115184434, 4.66002107074911457044767329640, 5.52935531655825512705437523890, 6.45354592331089491291581107900, 7.10097507023005863153467861656, 7.70580819100260057630002073703, 8.620925990908311200375000500775

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