Properties

Label 2-2960-740.187-c0-0-0
Degree $2$
Conductor $2960$
Sign $0.976 - 0.217i$
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  + (−0.939 − 0.342i)5-s + (−0.984 + 0.173i)9-s + (0.984 + 0.173i)13-s + (0.118 + 0.673i)17-s + (0.766 + 0.642i)25-s + (1.10 + 0.296i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (0.984 + 0.173i)45-s + (0.642 + 0.766i)49-s + (0.469 − 0.218i)53-s + (1.34 − 0.939i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (−0.984 + 0.173i)9-s + (0.984 + 0.173i)13-s + (0.118 + 0.673i)17-s + (0.766 + 0.642i)25-s + (1.10 + 0.296i)29-s + (0.173 − 0.984i)37-s + (0.673 + 0.118i)41-s + (0.984 + 0.173i)45-s + (0.642 + 0.766i)49-s + (0.469 − 0.218i)53-s + (1.34 − 0.939i)61-s + (−0.866 − 0.5i)65-s + (1.36 + 1.36i)73-s + (0.939 − 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.976 - 0.217i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (927, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :0),\ 0.976 - 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9740492854\)
\(L(\frac12)\) \(\approx\) \(0.9740492854\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
good3 \( 1 + (0.984 - 0.173i)T^{2} \)
7 \( 1 + (-0.642 - 0.766i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.984 - 0.173i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (0.984 - 0.173i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.10 - 0.296i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.469 + 0.218i)T + (0.642 - 0.766i)T^{2} \)
59 \( 1 + (0.642 - 0.766i)T^{2} \)
61 \( 1 + (-1.34 + 0.939i)T + (0.342 - 0.939i)T^{2} \)
67 \( 1 + (0.642 + 0.766i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + (-0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.342 - 0.939i)T^{2} \)
89 \( 1 + (0.766 - 0.357i)T + (0.642 - 0.766i)T^{2} \)
97 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.649549042880166627461202220315, −8.379402158007383193385534885116, −7.60980863131119412655429196115, −6.67502455181599489604750323692, −5.88600092770903266985619119445, −5.11021915254215613047930090498, −4.11837834371221374156900199153, −3.50004131091887853840304646324, −2.45055100381118556080234964447, −0.994729876300136376777135210251, 0.819524518260965400182535061946, 2.54783586090650312848475393402, 3.27601664271092639459299585828, 4.07049949025508409957748414499, 5.02104395878714992829452478322, 5.93529312831385438708703892472, 6.64541657229260222870525569930, 7.43746804229344455219227677759, 8.332797391665756482620581533815, 8.596379503304978747415972658998

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