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Properties

Label 1-104-104.77-r0-0-0
Degree $1$
Conductor $104$
Sign $1$
Analytic cond. $0.482973$
Root an. cond. $0.482973$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s − 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 41-s − 43-s + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s − 57-s + 59-s + ⋯
L(s)  = 1  − 3-s + 5-s − 7-s + 9-s + 11-s − 15-s + 17-s + 19-s + 21-s + 23-s + 25-s − 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 41-s − 43-s + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s − 57-s + 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.482973\)
Root analytic conductor: \(0.482973\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{104} (77, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8750666998\)
\(L(\frac12)\) \(\approx\) \(0.8750666998\)
\(L(1)\) \(\approx\) \(0.9070129404\)
\(L(1)\) \(\approx\) \(0.9070129404\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.5096353772718546788974888875, −28.88885598313403019925322001465, −27.924856171979981876576125229046, −26.76288212088677660500730805150, −25.48102919548405567519246130769, −24.716032362294075183647369381635, −23.36399902596304044839772546573, −22.35753526992902581834247197095, −21.81015070031660427214621862639, −20.49265498993951683861831252313, −19.03167445007081970446097952905, −18.09525212087343326583733874672, −16.86992553228994345386835971299, −16.4414791464035576218382167085, −14.82791241339491441147557818157, −13.43836036271794521901736933929, −12.52648009351914427989300321012, −11.33239468034698603368518572027, −9.97440811482177937703459310240, −9.32988042748789355552564433362, −7.12706752621834119887642391507, −6.169822844033692662531572588, −5.20201547865457785022578707691, −3.41481777469765525985532567540, −1.370583964578187003116024339483, 1.370583964578187003116024339483, 3.41481777469765525985532567540, 5.20201547865457785022578707691, 6.169822844033692662531572588, 7.12706752621834119887642391507, 9.32988042748789355552564433362, 9.97440811482177937703459310240, 11.33239468034698603368518572027, 12.52648009351914427989300321012, 13.43836036271794521901736933929, 14.82791241339491441147557818157, 16.4414791464035576218382167085, 16.86992553228994345386835971299, 18.09525212087343326583733874672, 19.03167445007081970446097952905, 20.49265498993951683861831252313, 21.81015070031660427214621862639, 22.35753526992902581834247197095, 23.36399902596304044839772546573, 24.716032362294075183647369381635, 25.48102919548405567519246130769, 26.76288212088677660500730805150, 27.924856171979981876576125229046, 28.88885598313403019925322001465, 29.5096353772718546788974888875

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