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Properties

Label 2-104-104.69-c1-0-7
Degree $2$
Conductor $104$
Sign $0.494 + 0.869i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (0.366 − 1.36i)2-s + (0.633 + 0.366i)3-s + (−1.73 − i)4-s + 3.73·5-s + (0.732 − 0.732i)6-s + (−3 + 1.73i)7-s + (−2 + 1.99i)8-s + (−1.23 − 2.13i)9-s + (1.36 − 5.09i)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 + (1.99 + 3.46i)16-s + (0.232 + 0.401i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.366 + 0.211i)3-s + (−0.866 − 0.5i)4-s + 1.66·5-s + (0.298 − 0.298i)6-s + (−1.13 + 0.654i)7-s + (−0.707 + 0.707i)8-s + (−0.410 − 0.711i)9-s + (0.431 − 1.61i)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 + (0.499 + 0.866i)16-s + (0.0562 + 0.0974i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.494 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10059 - 0.640144i\)
\(L(\frac12)\) \(\approx\) \(1.10059 - 0.640144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
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

−13.68487214625966252566828586197, −12.58792125577513754843320534867, −11.69272639658502696953682085000, −10.06997274265500595141798590178, −9.512756683316004384601837538598, −8.908559251459995551990744801681, −6.30479423701509434950843452068, −5.55034809440366847308401048876, −3.50044679182669776362616364966, −2.22246397514802082514205269828, 2.79563494400777199708628505245, 4.88613071800989277155250540128, 6.14559647065851798915674748456, 7.00963515152245142485097021304, 8.408860199936460454095466567285, 9.702696483346020277062097088550, 10.20821305767706023121690391563, 12.55157854028107888660076389757, 13.23259971866806694876623277241, 13.98495319258192499784782143244

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