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Properties

Label 2-104-104.101-c1-0-11
Degree $2$
Conductor $104$
Sign $-0.869 - 0.494i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.633 + 0.366i)3-s + 2i·4-s − 3.73·5-s + (1 + 0.267i)6-s + (−3 − 1.73i)7-s + (2 − 2i)8-s + (−1.23 + 2.13i)9-s + (3.73 + 3.73i)10-s + (−1 − 1.73i)11-s + (−0.732 − 1.26i)12-s + (2.59 + 2.5i)13-s + (1.26 + 4.73i)14-s + (2.36 − 1.36i)15-s − 4·16-s + (0.232 − 0.401i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.366 + 0.211i)3-s + i·4-s − 1.66·5-s + (0.408 + 0.109i)6-s + (−1.13 − 0.654i)7-s + (0.707 − 0.707i)8-s + (−0.410 + 0.711i)9-s + (1.18 + 1.18i)10-s + (−0.301 − 0.522i)11-s + (−0.211 − 0.366i)12-s + (0.720 + 0.693i)13-s + (0.338 + 1.26i)14-s + (0.610 − 0.352i)15-s − 16-s + (0.0562 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.869 - 0.494i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ -0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good3 \( 1 + (0.633 - 0.366i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.633 - 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.09 + 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.59 - 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73iT - 31T^{2} \)
37 \( 1 + (2.13 + 3.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.96 - 4.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.19 + 1.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 - 3.92iT - 53T^{2} \)
59 \( 1 + (0.267 - 0.464i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 + 6.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.02 - 4.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 1.46T + 83T^{2} \)
89 \( 1 + (-6.46 + 3.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.82843358379086968609241695444, −11.87422779914322198786002628128, −10.99008065912122407362118649739, −10.32069971646691621635120490554, −8.740548967189365823097484865780, −7.890101517039449672594169451293, −6.67002848047013952318864383646, −4.31439426193152263666412656159, −3.28122465418330377340162477398, 0, 3.58033679028000759983194361425, 5.57173060992550828612275010275, 6.70241883468960821824616526454, 7.79482891601130147892503056776, 8.786923276720999196671280305504, 9.959275773227621639853646702370, 11.34658401651644295602498313189, 12.09155432331919422560021806903, 13.27459514992018053723281735713

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