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Properties

Label 2-3240-9.7-c1-0-12
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 + (3 + 5.19i)13-s − 2·17-s + 4·19-s + (4 + 6.92i)23-s + (−0.499 + 0.866i)25-s + (3 − 5.19i)29-s − 3.99·35-s − 6·37-s + (−5 − 8.66i)41-s + (2 − 3.46i)43-s + (−4 + 6.92i)47-s + (−4.49 − 7.79i)49-s + 10·53-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.755 + 1.30i)7-s + (0.832 + 1.44i)13-s − 0.485·17-s + 0.917·19-s + (0.834 + 1.44i)23-s + (−0.0999 + 0.173i)25-s + (0.557 − 0.964i)29-s − 0.676·35-s − 0.986·37-s + (−0.780 − 1.35i)41-s + (0.304 − 0.528i)43-s + (−0.583 + 1.01i)47-s + (−0.642 − 1.11i)49-s + 1.37·53-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: $\chi_{3240} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437498415\)
\(L(\frac12)\) \(\approx\) \(1.437498415\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.021955830971184768341467848600, −8.452228517906154571917514152264, −7.18679268421234674976718138921, −6.76219500454870847235761494505, −5.84601886811357337710145357512, −5.41736802813952700440168700397, −4.18663179821168854929292902141, −3.31469922107564753550298921797, −2.49161650569277382209807450462, −1.51906368350161753229003418168, 0.46679702817540019180239439020, 1.31698527142592698187063889768, 2.96010191501866059387315449448, 3.46821517886326971390121910378, 4.52978864261540922905526709202, 5.23716873636418563500879379326, 6.22804122221845090099019553497, 6.84137355146072691426082589887, 7.56813137299552079105200355009, 8.456353585640741352048857911441

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