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Properties

Label 2-3120-195.194-c0-0-5
Degree $2$
Conductor $3120$
Sign $0.707 + 0.707i$
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  − i·3-s + (0.707 + 0.707i)5-s − 9-s + 1.41·11-s − i·13-s + (0.707 − 0.707i)15-s + 1.00i·25-s + i·27-s − 1.41i·33-s − 39-s + 1.41·41-s + (−0.707 − 0.707i)45-s − 1.41i·47-s − 49-s + (1.00 + 1.00i)55-s + ⋯
L(s)  = 1  − i·3-s + (0.707 + 0.707i)5-s − 9-s + 1.41·11-s − i·13-s + (0.707 − 0.707i)15-s + 1.00i·25-s + i·27-s − 1.41i·33-s − 39-s + 1.41·41-s + (−0.707 − 0.707i)45-s − 1.41i·47-s − 49-s + (1.00 + 1.00i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496924230\)
\(L(\frac12)\) \(\approx\) \(1.496924230\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + iT \)
good7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.754633155535384756692357859731, −7.923540848893777198031620159253, −7.14836188566454408092073151975, −6.56170845353891399188802753041, −5.94779611729239151603374940923, −5.24330485548631865963137047763, −3.85967824181435793043587435044, −2.99868932822023694747026337560, −2.11359202764148365924145596105, −1.11118612169967912673604419407, 1.30931509903786147446137389439, 2.42822180340956885064004544794, 3.65841670227267768040229612962, 4.36957199833053953136693198093, 4.94014655909173857624445274157, 6.02082974387755687485925580192, 6.36986941926342727327050539704, 7.52790894450817661516704640978, 8.662748453064049680821350372640, 9.019677244089865123146113846484

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