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Properties

Label 2-240-80.3-c1-0-6
Degree $2$
Conductor $240$
Sign $0.649 - 0.760i$
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  + (−1.32 + 0.489i)2-s + 3-s + (1.52 − 1.29i)4-s + (2.06 + 0.849i)5-s + (−1.32 + 0.489i)6-s + (−2.08 + 2.08i)7-s + (−1.38 + 2.46i)8-s + 9-s + (−3.16 − 0.114i)10-s + (3.33 + 3.33i)11-s + (1.52 − 1.29i)12-s − 6.13i·13-s + (1.74 − 3.77i)14-s + (2.06 + 0.849i)15-s + (0.623 − 3.95i)16-s + (−2.33 + 2.33i)17-s + ⋯
L(s)  = 1  + (−0.938 + 0.346i)2-s + 0.577·3-s + (0.760 − 0.649i)4-s + (0.924 + 0.380i)5-s + (−0.541 + 0.199i)6-s + (−0.786 + 0.786i)7-s + (−0.488 + 0.872i)8-s + 0.333·9-s + (−0.999 − 0.0363i)10-s + (1.00 + 1.00i)11-s + (0.438 − 0.375i)12-s − 1.70i·13-s + (0.465 − 1.00i)14-s + (0.534 + 0.219i)15-s + (0.155 − 0.987i)16-s + (−0.565 + 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00044 + 0.460877i\)
\(L(\frac12)\) \(\approx\) \(1.00044 + 0.460877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.489i)T \)
3 \( 1 - T \)
5 \( 1 + (-2.06 - 0.849i)T \)
good7 \( 1 + (2.08 - 2.08i)T - 7iT^{2} \)
11 \( 1 + (-3.33 - 3.33i)T + 11iT^{2} \)
13 \( 1 + 6.13iT - 13T^{2} \)
17 \( 1 + (2.33 - 2.33i)T - 17iT^{2} \)
19 \( 1 + (-0.834 - 0.834i)T + 19iT^{2} \)
23 \( 1 + (-2.95 - 2.95i)T + 23iT^{2} \)
29 \( 1 + (0.576 - 0.576i)T - 29iT^{2} \)
31 \( 1 + 2.62iT - 31T^{2} \)
37 \( 1 + 2.07iT - 37T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 - 5.16iT - 43T^{2} \)
47 \( 1 + (8.65 + 8.65i)T + 47iT^{2} \)
53 \( 1 + 1.58T + 53T^{2} \)
59 \( 1 + (-2.32 + 2.32i)T - 59iT^{2} \)
61 \( 1 + (7.22 + 7.22i)T + 61iT^{2} \)
67 \( 1 - 0.885iT - 67T^{2} \)
71 \( 1 - 2.56T + 71T^{2} \)
73 \( 1 + (-7.35 + 7.35i)T - 73iT^{2} \)
79 \( 1 + 7.72T + 79T^{2} \)
83 \( 1 + 8.67T + 83T^{2} \)
89 \( 1 + 8.70T + 89T^{2} \)
97 \( 1 + (-11.9 + 11.9i)T - 97iT^{2} \)
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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

−12.34585631547687463503284952008, −10.93487637695770640882327540244, −9.920823569404718527073105269732, −9.453496805387010998833206547471, −8.553713333205982994225444874842, −7.30996551265207798303304383932, −6.38167906388547775384801901767, −5.44422474354075592098418854172, −3.14393138082321034213637388165, −1.90385095972494446443205662927, 1.34121579305514309465991624559, 2.91738582877832170663101134183, 4.26388359639038852889129465727, 6.46027780142952292868789558076, 6.88080789918416947041207575098, 8.491891682587120506298228357400, 9.257935326313301143764450586412, 9.693587395310088299805632601085, 10.89614270928576968639953146660, 11.79766132095336052901852632587

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