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Properties

Label 2-104-8.3-c2-0-20
Degree $2$
Conductor $104$
Sign $-0.942 - 0.334i$
Analytic cond. $2.83379$
Root an. cond. $1.68338$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 1.19i)2-s − 1.48·3-s + (1.16 + 3.82i)4-s − 7.88i·5-s + (2.39 + 1.77i)6-s + 5.80i·7-s + (2.67 − 7.53i)8-s − 6.78·9-s + (−9.38 + 12.6i)10-s − 15.7·11-s + (−1.73 − 5.69i)12-s + 3.60i·13-s + (6.91 − 9.33i)14-s + 11.7i·15-s + (−13.2 + 8.92i)16-s − 28.1·17-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)2-s − 0.496·3-s + (0.291 + 0.956i)4-s − 1.57i·5-s + (0.398 + 0.295i)6-s + 0.829i·7-s + (0.334 − 0.942i)8-s − 0.753·9-s + (−0.938 + 1.26i)10-s − 1.43·11-s + (−0.144 − 0.474i)12-s + 0.277i·13-s + (0.493 − 0.666i)14-s + 0.783i·15-s + (−0.829 + 0.557i)16-s − 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.942 - 0.334i$
Analytic conductor: \(2.83379\)
Root analytic conductor: \(1.68338\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1),\ -0.942 - 0.334i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0352911 + 0.204678i\)
\(L(\frac12)\) \(\approx\) \(0.0352911 + 0.204678i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.60 + 1.19i)T \)
13 \( 1 - 3.60iT \)
good3 \( 1 + 1.48T + 9T^{2} \)
5 \( 1 + 7.88iT - 25T^{2} \)
7 \( 1 - 5.80iT - 49T^{2} \)
11 \( 1 + 15.7T + 121T^{2} \)
17 \( 1 + 28.1T + 289T^{2} \)
19 \( 1 - 12.5T + 361T^{2} \)
23 \( 1 - 4.58iT - 529T^{2} \)
29 \( 1 + 33.0iT - 841T^{2} \)
31 \( 1 + 32.1iT - 961T^{2} \)
37 \( 1 + 37.4iT - 1.36e3T^{2} \)
41 \( 1 - 39.7T + 1.68e3T^{2} \)
43 \( 1 + 57.5T + 1.84e3T^{2} \)
47 \( 1 - 61.1iT - 2.20e3T^{2} \)
53 \( 1 - 18.5iT - 2.80e3T^{2} \)
59 \( 1 + 35.6T + 3.48e3T^{2} \)
61 \( 1 + 69.7iT - 3.72e3T^{2} \)
67 \( 1 - 76.7T + 4.48e3T^{2} \)
71 \( 1 + 11.0iT - 5.04e3T^{2} \)
73 \( 1 + 79.1T + 5.32e3T^{2} \)
79 \( 1 + 94.0iT - 6.24e3T^{2} \)
83 \( 1 + 116.T + 6.88e3T^{2} \)
89 \( 1 - 58.8T + 7.92e3T^{2} \)
97 \( 1 - 26.8T + 9.40e3T^{2} \)
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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

−12.73368736707989196368670423170, −11.79813673252700507731208919445, −11.02260248542054866393214654137, −9.508952494424078243530696387403, −8.749504660239800707651127477209, −7.86777213442934592151820946325, −5.86979601066309190448894528556, −4.65308096815937054711237237753, −2.36708812703496561674574013555, −0.19643799733607831842972183981, 2.78101305241856275734155602180, 5.22106975866334247618627351597, 6.56868951641313200303454349667, 7.26822403732383625427005729709, 8.503532054554879958215933149810, 10.22779872052754477432047179834, 10.69918343589881121286953782531, 11.42435272352872055721679706470, 13.43218499121055648675974507855, 14.30699259336959349933290199868

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