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Properties

Label 2-2960-740.327-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.988 + 0.148i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + 5-s + i·9-s − 2i·13-s + 25-s + (1 − i)29-s + 37-s + i·45-s + i·49-s + (−1 + i)53-s + (1 + i)61-s − 2i·65-s + (−1 − i)73-s − 81-s + (−1 + i)89-s + 2·97-s + ⋯
L(s)  = 1  + 5-s + i·9-s − 2i·13-s + 25-s + (1 − i)29-s + 37-s + i·45-s + i·49-s + (−1 + i)53-s + (1 + i)61-s − 2i·65-s + (−1 − i)73-s − 81-s + (−1 + i)89-s + 2·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (1807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520859467\)
\(L(\frac12)\) \(\approx\) \(1.520859467\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
37 \( 1 - T \)
good3 \( 1 - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−8.913067598529495488702094163069, −7.995737472845140081692412820653, −7.67040064438679314514119852193, −6.47376079546832560673368628954, −5.78028760957398732186724775059, −5.20662776530580697136081023284, −4.37015151238388350983553211857, −2.96644912324728096945516213202, −2.45970070691212981498848123866, −1.14820562126073964932029727565, 1.31579764839845224557079622792, 2.23163055592869945829565136046, 3.32047058723932312643792262325, 4.29669721301503063957824826984, 5.06694596616062340701323294831, 6.11218130779767829218663894678, 6.61266789467305997149464159372, 7.14994014115242988470709014104, 8.496464401155093086501324871858, 8.981561352064478990295532479198

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