Properties

Label 2-240-80.29-c1-0-19
Degree $2$
Conductor $240$
Sign $0.598 + 0.801i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (1.20 − 0.742i)2-s + (0.707 − 0.707i)3-s + (0.898 − 1.78i)4-s + (0.860 + 2.06i)5-s + (0.326 − 1.37i)6-s − 0.707·7-s + (−0.244 − 2.81i)8-s − 1.00i·9-s + (2.56 + 1.84i)10-s + (1.79 − 1.79i)11-s + (−0.628 − 1.89i)12-s + (−3.86 + 3.86i)13-s + (−0.851 + 0.524i)14-s + (2.06 + 0.850i)15-s + (−2.38 − 3.21i)16-s + 0.244i·17-s + ⋯
L(s)  = 1  + (0.851 − 0.524i)2-s + (0.408 − 0.408i)3-s + (0.449 − 0.893i)4-s + (0.384 + 0.922i)5-s + (0.133 − 0.561i)6-s − 0.267·7-s + (−0.0863 − 0.996i)8-s − 0.333i·9-s + (0.812 + 0.583i)10-s + (0.542 − 0.542i)11-s + (−0.181 − 0.548i)12-s + (−1.07 + 1.07i)13-s + (−0.227 + 0.140i)14-s + (0.533 + 0.219i)15-s + (−0.596 − 0.802i)16-s + 0.0593i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.598 + 0.801i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95818 - 0.981648i\)
\(L(\frac12)\) \(\approx\) \(1.95818 - 0.981648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.742i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.860 - 2.06i)T \)
good7 \( 1 + 0.707T + 7T^{2} \)
11 \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \)
13 \( 1 + (3.86 - 3.86i)T - 13iT^{2} \)
17 \( 1 - 0.244iT - 17T^{2} \)
19 \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + (4.89 + 4.89i)T + 29iT^{2} \)
31 \( 1 - 7.60T + 31T^{2} \)
37 \( 1 + (-8.47 - 8.47i)T + 37iT^{2} \)
41 \( 1 - 2.12iT - 41T^{2} \)
43 \( 1 + (-0.684 - 0.684i)T + 43iT^{2} \)
47 \( 1 - 4.47iT - 47T^{2} \)
53 \( 1 + (1.47 + 1.47i)T + 53iT^{2} \)
59 \( 1 + (-5.86 + 5.86i)T - 59iT^{2} \)
61 \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \)
67 \( 1 + (-7.85 + 7.85i)T - 67iT^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + 9.69T + 73T^{2} \)
79 \( 1 + 7.34T + 79T^{2} \)
83 \( 1 + (-6.80 + 6.80i)T - 83iT^{2} \)
89 \( 1 + 3.07iT - 89T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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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

−11.84870479957560637257502551978, −11.43898740238839909997633570974, −9.963426691601558360073673241155, −9.577324193967937897604825534149, −7.81763723707092330610099211055, −6.61802705399806848001133188289, −6.04129664314537904920120998604, −4.33826980465850332137698030332, −3.10392869833740293905464813253, −1.97647857477971439053557347247, 2.43876316184202812763620288744, 3.96075727441017176208927438165, 4.96946005159928655091882681518, 5.88703730405084705608471539798, 7.30305660168177527491317502248, 8.218284956892367780633333101623, 9.326849385626968390840312438271, 10.16582863126688907044232218822, 11.69633177584465970704693970983, 12.54316854076634477583151658302

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