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Properties

Label 2-104-13.3-c1-0-1
Degree $2$
Conductor $104$
Sign $0.859 + 0.511i$
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  + (−0.5 − 0.866i)3-s + 2·5-s + (0.5 − 0.866i)7-s + (1 − 1.73i)9-s + (−0.5 − 0.866i)11-s + (−1 + 3.46i)13-s + (−1 − 1.73i)15-s + (−1.5 + 2.59i)17-s + (−3.5 + 6.06i)19-s − 0.999·21-s + (−0.5 − 0.866i)23-s − 25-s − 5·27-s + (−1.5 − 2.59i)29-s + 8·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + 0.894·5-s + (0.188 − 0.327i)7-s + (0.333 − 0.577i)9-s + (−0.150 − 0.261i)11-s + (−0.277 + 0.960i)13-s + (−0.258 − 0.447i)15-s + (−0.363 + 0.630i)17-s + (−0.802 + 1.39i)19-s − 0.218·21-s + (−0.104 − 0.180i)23-s − 0.200·25-s − 0.962·27-s + (−0.278 − 0.482i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01646 - 0.279359i\)
\(L(\frac12)\) \(\approx\) \(1.01646 - 0.279359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.5 + 9.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.5 + 4.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.67338800469130127364851304743, −12.68662080982883412099499869639, −11.76004934409619127987703453911, −10.44024030447475025031996535876, −9.525959149125548442170278174527, −8.167516996140399180117337750497, −6.71491554184052603124923928704, −5.92135947979335187446502644488, −4.14345910193158530011875188027, −1.83228794018555686871150659946, 2.43897965716384504950270600961, 4.67460463112019886015593311308, 5.60900411333539946589243347304, 7.09637732839689137909266446190, 8.574120347462755734274642843815, 9.816946494193511401443504654190, 10.52085077904631868882676399988, 11.66850121768190590489068512824, 13.06098010197014093996098963555, 13.67721496264820850969332970863

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