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Properties

Label 2-960-80.3-c1-0-14
Degree $2$
Conductor $960$
Sign $0.0290 + 0.999i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−2.13 + 0.658i)5-s + (3.54 − 3.54i)7-s + 9-s + (0.707 + 0.707i)11-s + 1.18i·13-s + (2.13 − 0.658i)15-s + (−2.63 + 2.63i)17-s + (−5.21 − 5.21i)19-s + (−3.54 + 3.54i)21-s + (1.86 + 1.86i)23-s + (4.13 − 2.81i)25-s − 27-s + (2.17 − 2.17i)29-s − 2.39i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.955 + 0.294i)5-s + (1.34 − 1.34i)7-s + 0.333·9-s + (0.213 + 0.213i)11-s + 0.329i·13-s + (0.551 − 0.170i)15-s + (−0.639 + 0.639i)17-s + (−1.19 − 1.19i)19-s + (−0.774 + 0.774i)21-s + (0.388 + 0.388i)23-s + (0.826 − 0.562i)25-s − 0.192·27-s + (0.403 − 0.403i)29-s − 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0290 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0290 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0290 + 0.999i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.0290 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702965 - 0.682808i\)
\(L(\frac12)\) \(\approx\) \(0.702965 - 0.682808i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (2.13 - 0.658i)T \)
good7 \( 1 + (-3.54 + 3.54i)T - 7iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + 11iT^{2} \)
13 \( 1 - 1.18iT - 13T^{2} \)
17 \( 1 + (2.63 - 2.63i)T - 17iT^{2} \)
19 \( 1 + (5.21 + 5.21i)T + 19iT^{2} \)
23 \( 1 + (-1.86 - 1.86i)T + 23iT^{2} \)
29 \( 1 + (-2.17 + 2.17i)T - 29iT^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + 0.910iT - 37T^{2} \)
41 \( 1 + 8.26iT - 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 + (5.06 + 5.06i)T + 47iT^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 + (-10.2 + 10.2i)T - 59iT^{2} \)
61 \( 1 + (-4.49 - 4.49i)T + 61iT^{2} \)
67 \( 1 - 1.27iT - 67T^{2} \)
71 \( 1 - 3.56T + 71T^{2} \)
73 \( 1 + (2.47 - 2.47i)T - 73iT^{2} \)
79 \( 1 + 3.89T + 79T^{2} \)
83 \( 1 + 9.99T + 83T^{2} \)
89 \( 1 + 5.16T + 89T^{2} \)
97 \( 1 + (6.87 - 6.87i)T - 97iT^{2} \)
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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

−10.18332605387011223852595550319, −8.745667376144678671621416218053, −8.127251241270472680923325684402, −6.96847897032918366178493866647, −6.91623294398779808820450188192, −5.25569458195805703202851457924, −4.29585340501749025209866462823, −3.95559240918277387577727644100, −2.03793537146418337708930142386, −0.53321099982145651262334543614, 1.39933812597711903030519505507, 2.79321338477246862840702057915, 4.31229645912697937378101752773, 4.90884346647219467349208546789, 5.81028351176249481920489242236, 6.78338416934721774952343017891, 8.036219004958736205415490066231, 8.361733220049740518000602141114, 9.201018751058938840151545096614, 10.46022950821491630864294424224

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