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Properties

Label 1-2960-2960.2309-r1-0-0
Degree $1$
Conductor $2960$
Sign $-0.984 + 0.176i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  + (−0.642 − 0.766i)3-s + (−0.939 − 0.342i)7-s + (−0.173 + 0.984i)9-s + (0.866 + 0.5i)11-s + (0.173 + 0.984i)13-s + (0.984 + 0.173i)17-s + (−0.766 + 0.642i)19-s + (0.342 + 0.939i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s + i·31-s + (−0.173 − 0.984i)33-s + (0.642 − 0.766i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (−0.939 − 0.342i)7-s + (−0.173 + 0.984i)9-s + (0.866 + 0.5i)11-s + (0.173 + 0.984i)13-s + (0.984 + 0.173i)17-s + (−0.766 + 0.642i)19-s + (0.342 + 0.939i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s + i·31-s + (−0.173 − 0.984i)33-s + (0.642 − 0.766i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ -0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06810432233 + 0.7640042815i\)
\(L(\frac12)\) \(\approx\) \(0.06810432233 + 0.7640042815i\)
\(L(1)\) \(\approx\) \(0.7324218294 + 0.07206395925i\)
\(L(1)\) \(\approx\) \(0.7324218294 + 0.07206395925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.342 + 0.939i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.64219955468706500017026485845, −17.76410303010679113174786844546, −16.97715314062190717465435015063, −16.5900149893622835429014383277, −15.81468217706973571175359210642, −15.211910555452772002426289674423, −14.60967416701798347053954898536, −13.59375072785284138914107436321, −12.85389357459302505941399036410, −12.05254697615346198938521773499, −11.59721838564628220600961593903, −10.59973003564119737135203671050, −10.10311610606917356102030636826, −9.36449389748371008675946002760, −8.74613688483104940430904060490, −7.80541354087656614831831579369, −6.68419925408087112629458719409, −6.0495028880906204285332446427, −5.613270718139844913516691287741, −4.58709445249150720082286787994, −3.67610093902682835275321064007, −3.25307118174579992272168991340, −2.106395239835638075173301598011, −0.608820401629850727795085174191, −0.213270811214901114743560110929, 1.24966096561935576148481063644, 1.604732505282365122410853587365, 2.82818014717597680229707833464, 3.82703900385314314075734020044, 4.48308755445446035428067158714, 5.67291599157069595038627448460, 6.21264646139194130735748878059, 6.91414237391757025336031740409, 7.43711471336033503678988457214, 8.42239330365565028630549791793, 9.2915932341616213887173859632, 10.062684663014049023198051351136, 10.70739694444832714138054065128, 11.68985973827794688467781736577, 12.20907923945882717748475875103, 12.73312980878129521259407874742, 13.59042842058094925250851325312, 14.20442155366112321420987487899, 14.90434310635311868777767585269, 16.18080451191250349435938325071, 16.47722565153187982523028856850, 17.080372896350408501639985946234, 17.84019798670742995633332405353, 18.64749096239591844941853097810, 19.18591450732723807704988186299

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