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Properties

Label 2-960-120.29-c2-0-14
Degree $2$
Conductor $960$
Sign $0.268 - 0.963i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 − 0.751i)3-s + (1.38 − 4.80i)5-s − 3.03i·7-s + (7.86 + 4.36i)9-s − 11.4·11-s − 16.9·13-s + (−7.63 + 12.9i)15-s + 18.1·17-s + 16.9i·19-s + (−2.28 + 8.82i)21-s − 22.3·23-s + (−21.1 − 13.3i)25-s + (−19.5 − 18.5i)27-s + 17.8·29-s − 16.9·31-s + ⋯
L(s)  = 1  + (−0.968 − 0.250i)3-s + (0.277 − 0.960i)5-s − 0.434i·7-s + (0.874 + 0.485i)9-s − 1.04·11-s − 1.30·13-s + (−0.509 + 0.860i)15-s + 1.06·17-s + 0.891i·19-s + (−0.108 + 0.420i)21-s − 0.971·23-s + (−0.846 − 0.532i)25-s + (−0.724 − 0.688i)27-s + 0.617·29-s − 0.547·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.268 - 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4713188866\)
\(L(\frac12)\) \(\approx\) \(0.4713188866\)
\(L(2)\) 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 \( 1 + (2.90 + 0.751i)T \)
5 \( 1 + (-1.38 + 4.80i)T \)
good7 \( 1 + 3.03iT - 49T^{2} \)
11 \( 1 + 11.4T + 121T^{2} \)
13 \( 1 + 16.9T + 169T^{2} \)
17 \( 1 - 18.1T + 289T^{2} \)
19 \( 1 - 16.9iT - 361T^{2} \)
23 \( 1 + 22.3T + 529T^{2} \)
29 \( 1 - 17.8T + 841T^{2} \)
31 \( 1 + 16.9T + 961T^{2} \)
37 \( 1 + 1.21T + 1.36e3T^{2} \)
41 \( 1 - 42.0iT - 1.68e3T^{2} \)
43 \( 1 + 1.98T + 1.84e3T^{2} \)
47 \( 1 - 52.4T + 2.20e3T^{2} \)
53 \( 1 + 8.58iT - 2.80e3T^{2} \)
59 \( 1 + 93.0T + 3.48e3T^{2} \)
61 \( 1 + 80.4iT - 3.72e3T^{2} \)
67 \( 1 - 38.1T + 4.48e3T^{2} \)
71 \( 1 - 109. iT - 5.04e3T^{2} \)
73 \( 1 + 28.2iT - 5.32e3T^{2} \)
79 \( 1 - 56.1T + 6.24e3T^{2} \)
83 \( 1 - 126. iT - 6.88e3T^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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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

−10.10877257030545143440543422256, −9.442034074267204993883251922194, −7.88224258100698161662981283230, −7.76907368555419802085619066503, −6.46306635529314442557822546836, −5.47395292427245202075008252582, −5.03115001544263331771531714296, −3.98953512801005318237134690720, −2.27851421479339766060659483385, −1.02294620695690844620249567348, 0.19644289087968878829019938654, 2.15182207287615211781095154284, 3.14682285301843141298900380738, 4.52788098836495983836556030781, 5.46211633795399625292348013789, 6.01813182643180187473044905827, 7.19933771481035704998999062053, 7.59409319368278727231642131514, 9.055189514820565516595111199522, 10.04862774359891014970544633423

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