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Properties

Label 2-3120-195.47-c0-0-1
Degree $2$
Conductor $3120$
Sign $-0.966 + 0.256i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s − 7-s + 1.00i·9-s + (0.707 + 0.707i)11-s − 13-s − 1.00·15-s + (−0.707 − 0.707i)17-s + (0.707 + 0.707i)21-s + (0.707 − 0.707i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41·29-s + (−1 − i)31-s − 1.00i·33-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s − 7-s + 1.00i·9-s + (0.707 + 0.707i)11-s − 13-s − 1.00·15-s + (−0.707 − 0.707i)17-s + (0.707 + 0.707i)21-s + (0.707 − 0.707i)23-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41·29-s + (−1 − i)31-s − 1.00i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5115529550\)
\(L(\frac12)\) \(\approx\) \(0.5115529550\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 + T + T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.752644247581912086093117104842, −7.40458004294565349874252029802, −7.03728597173755476697821949029, −6.32697998576902313861606649800, −5.47892598912793289791540686145, −4.91514188348048934112656198724, −3.93607739897995806918283607364, −2.47986719254794958525886354967, −1.81520637487947104283643524059, −0.32081805722866647974168627071, 1.66858616700135375308837788054, 3.11590657312081258084827797003, 3.51719742917936278941038965879, 4.66325617406163545462772689480, 5.56583500737394454925315134791, 6.18173743995387610877059556166, 6.73644529695289951595421468610, 7.44313521277167156968755966782, 8.896025686616272180167671689902, 9.335716281562271813015238943746

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