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Properties

Label 2-1638-39.5-c1-0-8
Degree $2$
Conductor $1638$
Sign $-0.194 - 0.980i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.98 − 1.98i)5-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.81i·10-s + (−0.872 + 0.872i)11-s + (−3.13 + 1.78i)13-s − 1.00i·14-s − 1.00·16-s + 7.84·17-s + (−1.52 + 1.52i)19-s + (1.98 − 1.98i)20-s − 1.23·22-s + 6.35·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.889 − 0.889i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.889i·10-s + (−0.262 + 0.262i)11-s + (−0.869 + 0.494i)13-s − 0.267i·14-s − 0.250·16-s + 1.90·17-s + (−0.349 + 0.349i)19-s + (0.444 − 0.444i)20-s − 0.262·22-s + 1.32·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358608594\)
\(L(\frac12)\) \(\approx\) \(1.358608594\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (3.13 - 1.78i)T \)
good5 \( 1 + (1.98 + 1.98i)T + 5iT^{2} \)
11 \( 1 + (0.872 - 0.872i)T - 11iT^{2} \)
17 \( 1 - 7.84T + 17T^{2} \)
19 \( 1 + (1.52 - 1.52i)T - 19iT^{2} \)
23 \( 1 - 6.35T + 23T^{2} \)
29 \( 1 - 9.83iT - 29T^{2} \)
31 \( 1 + (3.63 - 3.63i)T - 31iT^{2} \)
37 \( 1 + (0.000971 + 0.000971i)T + 37iT^{2} \)
41 \( 1 + (-0.421 - 0.421i)T + 41iT^{2} \)
43 \( 1 - 5.21iT - 43T^{2} \)
47 \( 1 + (5.01 - 5.01i)T - 47iT^{2} \)
53 \( 1 - 13.7iT - 53T^{2} \)
59 \( 1 + (-6.45 + 6.45i)T - 59iT^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 + (-0.957 + 0.957i)T - 67iT^{2} \)
71 \( 1 + (0.476 + 0.476i)T + 71iT^{2} \)
73 \( 1 + (0.580 + 0.580i)T + 73iT^{2} \)
79 \( 1 - 9.53T + 79T^{2} \)
83 \( 1 + (-6.68 - 6.68i)T + 83iT^{2} \)
89 \( 1 + (-4.11 + 4.11i)T - 89iT^{2} \)
97 \( 1 + (-2.58 + 2.58i)T - 97iT^{2} \)
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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

−9.395533239542976586670626866708, −8.724439114781024072667470372562, −7.73107563358167498269457459087, −7.41219969578256553795749628862, −6.43346674644364753849743933379, −5.18079318832664286538887563614, −4.87144291095597936742728215308, −3.79435090575424636532461585329, −3.00551631801733006525729648277, −1.23028624322960867965963226877, 0.48845714518404306521497012738, 2.35724423350953778870990064196, 3.18682850241868945612206269136, 3.79919222326032737169054518299, 5.03549981697513333245332610683, 5.72577217083299726951489596014, 6.79427282568601727998019453523, 7.53561482613970392447432822244, 8.192187824336319743752111000313, 9.413587702880520931196405946711

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