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Properties

Label 2-476-476.67-c0-0-0
Degree $2$
Conductor $476$
Sign $0.0633 - 0.997i$
Analytic cond. $0.237554$
Root an. cond. $0.487396$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s − 7-s + 0.999·8-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)21-s + 0.999·22-s + (1 + 1.73i)23-s + (−0.499 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s − 7-s + 0.999·8-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)21-s + 0.999·22-s + (1 + 1.73i)23-s + (−0.499 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(0.237554\)
Root analytic conductor: \(0.487396\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :0),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3743785309\)
\(L(\frac12)\) \(\approx\) \(0.3743785309\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.26246938350249775881440450465, −10.41918517941358334846528168972, −9.671652785089959592089114768578, −9.415898028367756444486774555359, −7.889587423899678767324600154886, −7.07674643995011102841156992952, −5.52278031065069754227797755913, −4.53342548396747871444269246282, −3.56463939227455072724241124916, −2.20929858287886171956204612219, 0.58359072686550270944538605702, 2.71756319310288640086755522984, 4.61746162742137428400038199163, 5.72079172578821552208646306030, 6.69650529050022840972059289102, 6.99044540008741661684377655260, 8.195791024364529130577245259835, 9.079886573931892356732382836531, 10.02319911829146260956479107459, 10.80836428889250303098667123006

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