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Properties

Label 2-240-80.3-c1-0-1
Degree $2$
Conductor $240$
Sign $-0.996 - 0.0798i$
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  + (−0.0770 + 1.41i)2-s + 3-s + (−1.98 − 0.217i)4-s + (−2.13 + 0.658i)5-s + (−0.0770 + 1.41i)6-s + (−3.54 + 3.54i)7-s + (0.460 − 2.79i)8-s + 9-s + (−0.765 − 3.06i)10-s + (−0.707 − 0.707i)11-s + (−1.98 − 0.217i)12-s + 1.18i·13-s + (−4.73 − 5.28i)14-s + (−2.13 + 0.658i)15-s + (3.90 + 0.865i)16-s + (−2.63 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.0544 + 0.998i)2-s + 0.577·3-s + (−0.994 − 0.108i)4-s + (−0.955 + 0.294i)5-s + (−0.0314 + 0.576i)6-s + (−1.34 + 1.34i)7-s + (0.162 − 0.986i)8-s + 0.333·9-s + (−0.242 − 0.970i)10-s + (−0.213 − 0.213i)11-s + (−0.573 − 0.0628i)12-s + 0.329i·13-s + (−1.26 − 1.41i)14-s + (−0.551 + 0.170i)15-s + (0.976 + 0.216i)16-s + (−0.639 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.996 - 0.0798i$
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: \((2,\ 240,\ (\ :1/2),\ -0.996 - 0.0798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0297794 + 0.744901i\)
\(L(\frac12)\) \(\approx\) \(0.0297794 + 0.744901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0770 - 1.41i)T \)
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

−12.60393953570839058699305300003, −12.02915244848368297496735077185, −10.35834024130781490955262797915, −9.347576755233487110439695701687, −8.603173017501750967370975132017, −7.71766289689317600644811454044, −6.59898261467646176637880711384, −5.72164992443394212526232026478, −4.09066397900404834375519269116, −2.99745562218334358860100560501, 0.58175488167602208106421502868, 2.97267014213894544179917428637, 3.78687757957935059674054295600, 4.85385470168996600027912368633, 6.98279921954791062169166727154, 7.76644350385258187475288765437, 9.033783429874416643914276659267, 9.769545719260021479981182734322, 10.68041745409888035621524120716, 11.65361869209411162039738352716

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