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Properties

Label 2-240-5.2-c2-0-0
Degree $2$
Conductor $240$
Sign $-0.484 - 0.874i$
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 + (−3.36 − 3.70i)5-s + (8.78 + 8.78i)7-s − 2.99i·9-s − 13.7·11-s + (−4.88 + 4.88i)13-s + (8.65 + 0.413i)15-s + (5.99 + 5.99i)17-s + 25.5i·19-s − 21.5·21-s + (−18.4 + 18.4i)23-s + (−2.38 + 24.8i)25-s + (3.67 + 3.67i)27-s + 37.2i·29-s − 31.6·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.672 − 0.740i)5-s + (1.25 + 1.25i)7-s − 0.333i·9-s − 1.24·11-s + (−0.376 + 0.376i)13-s + (0.576 + 0.0275i)15-s + (0.352 + 0.352i)17-s + 1.34i·19-s − 1.02·21-s + (−0.801 + 0.801i)23-s + (−0.0954 + 0.995i)25-s + (0.136 + 0.136i)27-s + 1.28i·29-s − 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ -0.484 - 0.874i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.438753 + 0.744555i\)
\(L(\frac12)\) \(\approx\) \(0.438753 + 0.744555i\)
\(L(2)\) 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 \( 1 + (3.36 + 3.70i)T \)
good7 \( 1 + (-8.78 - 8.78i)T + 49iT^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + (4.88 - 4.88i)T - 169iT^{2} \)
17 \( 1 + (-5.99 - 5.99i)T + 289iT^{2} \)
19 \( 1 - 25.5iT - 361T^{2} \)
23 \( 1 + (18.4 - 18.4i)T - 529iT^{2} \)
29 \( 1 - 37.2iT - 841T^{2} \)
31 \( 1 + 31.6T + 961T^{2} \)
37 \( 1 + (32.5 + 32.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 36.7T + 1.68e3T^{2} \)
43 \( 1 + (-24.5 + 24.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (20.4 + 20.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-33.5 + 33.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 7.54iT - 3.48e3T^{2} \)
61 \( 1 - 43.6T + 3.72e3T^{2} \)
67 \( 1 + (-60.1 - 60.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 18.1T + 5.04e3T^{2} \)
73 \( 1 + (-68.8 + 68.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 22.2iT - 6.24e3T^{2} \)
83 \( 1 + (-0.00221 + 0.00221i)T - 6.88e3iT^{2} \)
89 \( 1 + 77.1iT - 7.92e3T^{2} \)
97 \( 1 + (-29.4 - 29.4i)T + 9.40e3iT^{2} \)
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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

−12.18068947884060319655261976695, −11.39066739724860132857919559369, −10.45297203688039237205614614001, −9.181924547788601406513766904233, −8.295876160091933864089700770763, −7.56870774786185816091032715495, −5.49607639958100537950037950085, −5.25538483456982056085558099934, −3.84535539443648111771279090742, −1.89213658998693001076115331171, 0.47616650462900462278493896198, 2.53002479900066463083242202422, 4.22954568454200379604459161996, 5.22068201941008033847045217495, 6.80461879490119162293253304890, 7.66630954159683000512470500709, 8.099137840186890544166418387604, 10.04351968483199562741856927923, 10.88588832534291440670694978034, 11.30847238380998906772460822528

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