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Properties

Label 2-1920-1920.29-c0-0-1
Degree $2$
Conductor $1920$
Sign $0.941 + 0.336i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
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  + (−0.290 − 0.956i)2-s + (−0.0980 + 0.995i)3-s + (−0.831 + 0.555i)4-s + (0.881 − 0.471i)5-s + (0.980 − 0.195i)6-s + (0.773 + 0.634i)8-s + (−0.980 − 0.195i)9-s + (−0.707 − 0.707i)10-s + (−0.471 − 0.881i)12-s + (0.382 + 0.923i)15-s + (0.382 − 0.923i)16-s + (0.360 − 0.871i)17-s + (0.0980 + 0.995i)18-s + (0.448 + 1.47i)19-s + (−0.471 + 0.881i)20-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.0980 + 0.995i)3-s + (−0.831 + 0.555i)4-s + (0.881 − 0.471i)5-s + (0.980 − 0.195i)6-s + (0.773 + 0.634i)8-s + (−0.980 − 0.195i)9-s + (−0.707 − 0.707i)10-s + (−0.471 − 0.881i)12-s + (0.382 + 0.923i)15-s + (0.382 − 0.923i)16-s + (0.360 − 0.871i)17-s + (0.0980 + 0.995i)18-s + (0.448 + 1.47i)19-s + (−0.471 + 0.881i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.042752545\)
\(L(\frac12)\) \(\approx\) \(1.042752545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 + 0.956i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
5 \( 1 + (-0.881 + 0.471i)T \)
good7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (0.195 + 0.980i)T^{2} \)
13 \( 1 + (0.555 + 0.831i)T^{2} \)
17 \( 1 + (-0.360 + 0.871i)T + (-0.707 - 0.707i)T^{2} \)
19 \( 1 + (-0.448 - 1.47i)T + (-0.831 + 0.555i)T^{2} \)
23 \( 1 + (-1.59 + 1.06i)T + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (-0.195 + 0.980i)T^{2} \)
31 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
37 \( 1 + (-0.831 - 0.555i)T^{2} \)
41 \( 1 + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.980 - 0.195i)T^{2} \)
47 \( 1 + (-1.83 - 0.761i)T + (0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.301 - 0.247i)T + (0.195 + 0.980i)T^{2} \)
59 \( 1 + (0.555 - 0.831i)T^{2} \)
61 \( 1 + (-1.26 - 0.124i)T + (0.980 + 0.195i)T^{2} \)
67 \( 1 + (-0.980 - 0.195i)T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1.87 - 0.569i)T + (0.831 - 0.555i)T^{2} \)
89 \( 1 + (-0.382 - 0.923i)T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.425711979736080864640419452248, −8.943035410316241211890130633184, −8.240194235697336982575661041314, −7.08244962936253591615489374387, −5.73620210111394689392220766630, −5.16859284503079845222837618126, −4.39673322907077775785415046406, −3.35868147002420190777648456915, −2.55225176479048293361818252346, −1.18460624136507505350009319289, 1.13497901164359435096039850187, 2.30776179858977917781816227474, 3.54826147450151661954165612300, 5.11722866394464969010825842335, 5.60524600912997447131616002552, 6.40565218644316870527369658182, 7.16203717701029736901789038156, 7.50846344472881775201866301683, 8.677061245747683996535880742050, 9.163933745373036149093871193706

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