Properties

Label 2-240-5.3-c2-0-11
Degree $2$
Conductor $240$
Sign $-0.326 + 0.945i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (2.67 − 4.22i)5-s + (1.44 − 1.44i)7-s + 2.99i·9-s + 3.34·11-s + (−10.4 − 10.4i)13-s + (−8.44 + 1.89i)15-s + (−2.65 + 2.65i)17-s − 20.6i·19-s − 3.55·21-s + (−16.4 − 16.4i)23-s + (−10.6 − 22.5i)25-s + (3.67 − 3.67i)27-s − 0.853i·29-s + 18.6·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.534 − 0.844i)5-s + (0.207 − 0.207i)7-s + 0.333i·9-s + 0.304·11-s + (−0.803 − 0.803i)13-s + (−0.563 + 0.126i)15-s + (−0.155 + 0.155i)17-s − 1.08i·19-s − 0.169·21-s + (−0.715 − 0.715i)23-s + (−0.427 − 0.903i)25-s + (0.136 − 0.136i)27-s − 0.0294i·29-s + 0.603·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.745729 - 1.04645i\)
\(L(\frac12)\) \(\approx\) \(0.745729 - 1.04645i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-2.67 + 4.22i)T \)
good7 \( 1 + (-1.44 + 1.44i)T - 49iT^{2} \)
11 \( 1 - 3.34T + 121T^{2} \)
13 \( 1 + (10.4 + 10.4i)T + 169iT^{2} \)
17 \( 1 + (2.65 - 2.65i)T - 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (16.4 + 16.4i)T + 529iT^{2} \)
29 \( 1 + 0.853iT - 841T^{2} \)
31 \( 1 - 18.6T + 961T^{2} \)
37 \( 1 + (-38.0 + 38.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 28.6T + 1.68e3T^{2} \)
43 \( 1 + (22.4 + 22.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.7 - 19.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-28.6 - 28.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (-54.8 + 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 + 39.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-21.1 - 21.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-14.5 + 14.5i)T - 9.40e3iT^{2} \)
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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

−11.78987724319219440563783697016, −10.64701937581745607565138188004, −9.716946381634248002980129308392, −8.650913820800655940043348079769, −7.64847012972257220275329986923, −6.45615254347630906647724536268, −5.38049108409829782976410266800, −4.41780163088280891667467582333, −2.36466950258511962793369091954, −0.72005703561967498738586511131, 2.01644564885115703703541077465, 3.60130278412735954220263941563, 4.98019756758232554948747394497, 6.12679795728778196076819414974, 6.98709449270415728515025754828, 8.292774113657205071105400873135, 9.766354860293705483818065360785, 10.00053856575295287976577467713, 11.41444288605785667383034030957, 11.81551021456136947436880301712

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