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Properties

Label 2-2904-2904.1853-c0-0-0
Degree $2$
Conductor $2904$
Sign $-0.323 + 0.946i$
Analytic cond. $1.44928$
Root an. cond. $1.20386$
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  + (−0.736 + 0.676i)2-s + (−0.809 − 0.587i)3-s + (0.0855 − 0.996i)4-s + (−0.0556 + 1.94i)5-s + (0.993 − 0.113i)6-s + (−0.288 − 0.546i)7-s + (0.610 + 0.791i)8-s + (0.309 + 0.951i)9-s + (−1.27 − 1.47i)10-s + (−0.654 − 0.755i)11-s + (−0.654 + 0.755i)12-s + (0.582 + 0.207i)14-s + (1.18 − 1.54i)15-s + (−0.985 − 0.170i)16-s + (−0.870 − 0.491i)18-s + ⋯
L(s)  = 1  + (−0.736 + 0.676i)2-s + (−0.809 − 0.587i)3-s + (0.0855 − 0.996i)4-s + (−0.0556 + 1.94i)5-s + (0.993 − 0.113i)6-s + (−0.288 − 0.546i)7-s + (0.610 + 0.791i)8-s + (0.309 + 0.951i)9-s + (−1.27 − 1.47i)10-s + (−0.654 − 0.755i)11-s + (−0.654 + 0.755i)12-s + (0.582 + 0.207i)14-s + (1.18 − 1.54i)15-s + (−0.985 − 0.170i)16-s + (−0.870 − 0.491i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $-0.323 + 0.946i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (1853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ -0.323 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06444098151\)
\(L(\frac12)\) \(\approx\) \(0.06444098151\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.736 - 0.676i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
good5 \( 1 + (0.0556 - 1.94i)T + (-0.998 - 0.0570i)T^{2} \)
7 \( 1 + (0.288 + 0.546i)T + (-0.564 + 0.825i)T^{2} \)
13 \( 1 + (-0.696 - 0.717i)T^{2} \)
17 \( 1 + (0.870 - 0.491i)T^{2} \)
19 \( 1 + (0.736 - 0.676i)T^{2} \)
23 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (1.88 - 0.107i)T + (0.993 - 0.113i)T^{2} \)
31 \( 1 + (0.582 - 0.966i)T + (-0.466 - 0.884i)T^{2} \)
37 \( 1 + (-0.897 - 0.441i)T^{2} \)
41 \( 1 + (0.921 + 0.389i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.516 + 0.856i)T^{2} \)
53 \( 1 + (-1.94 + 0.336i)T + (0.941 - 0.336i)T^{2} \)
59 \( 1 + (0.100 + 0.498i)T + (-0.921 + 0.389i)T^{2} \)
61 \( 1 + (-0.0855 - 0.996i)T^{2} \)
67 \( 1 + (0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.198 + 0.980i)T^{2} \)
73 \( 1 + (1.39 + 1.43i)T + (-0.0285 + 0.999i)T^{2} \)
79 \( 1 + (-0.526 - 0.770i)T + (-0.362 + 0.931i)T^{2} \)
83 \( 1 + (1.56 + 0.768i)T + (0.610 + 0.791i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.0113 + 0.397i)T + (-0.998 + 0.0570i)T^{2} \)
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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

−8.455094629430900051015096145004, −7.59064567404646697439783626258, −7.16607671429922527945637433601, −6.70724649983605959784866273289, −5.85893077119867511632567960420, −5.41791777775014134168197795868, −3.92743832066054206737180422064, −2.83743921611862158063309993889, −1.80336573659554808955140799790, −0.05990269301258509192634415295, 1.26920372812347937287392529768, 2.35376487943324830452300821727, 3.88886630813103855686646517258, 4.34458706702690348647654680209, 5.34367630869397966573427072700, 5.75761482815395999724282024138, 7.15233413554940014933366518524, 7.890269796555495579819625333881, 8.768094162579489804596922379673, 9.299410275113010313143397977127

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