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Properties

Label 1-68-68.67-r1-0-0
Degree $1$
Conductor $68$
Sign $1$
Analytic cond. $7.30761$
Root an. cond. $7.30761$
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 + 13-s − 15-s − 19-s + 21-s + 23-s + 25-s + 27-s − 29-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 53-s − 55-s − 57-s − 59-s + ⋯
L(s)  = 1  + 3-s − 5-s + 7-s + 9-s + 11-s + 13-s − 15-s − 19-s + 21-s + 23-s + 25-s + 27-s − 29-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 53-s − 55-s − 57-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(7.30761\)
Root analytic conductor: \(7.30761\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{68} (67, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.257150991\)
\(L(\frac12)\) \(\approx\) \(2.257150991\)
\(L(1)\) \(\approx\) \(1.523896275\)
\(L(1)\) \(\approx\) \(1.523896275\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
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

−31.503054242471559662839066690681, −30.64361017458379875080787967044, −30.00622314422540471059637468851, −27.930257074485883916326100478097, −27.34728685226188592330143516621, −26.26407048904031119541032896702, −25.00913256532058415028349540196, −24.1081347175306710689896870347, −22.9677275319389216124008545142, −21.34987337037444395906175476939, −20.45339973830119126466827715147, −19.407958784681966179132735987477, −18.48114018538926536002419198386, −16.871677329236306449202018631409, −15.36621962366366626457960950807, −14.70532446521386293738987832131, −13.43116938602475132779912601451, −11.93841064459192424663389128855, −10.77544343173933321874544405283, −8.90092108221283287415487062543, −8.17544353214064951199909610337, −6.85500182822389129131087212594, −4.54440542129910715986061139417, −3.46477752661597635309065728952, −1.50090041534469822320623975300, 1.50090041534469822320623975300, 3.46477752661597635309065728952, 4.54440542129910715986061139417, 6.85500182822389129131087212594, 8.17544353214064951199909610337, 8.90092108221283287415487062543, 10.77544343173933321874544405283, 11.93841064459192424663389128855, 13.43116938602475132779912601451, 14.70532446521386293738987832131, 15.36621962366366626457960950807, 16.871677329236306449202018631409, 18.48114018538926536002419198386, 19.407958784681966179132735987477, 20.45339973830119126466827715147, 21.34987337037444395906175476939, 22.9677275319389216124008545142, 24.1081347175306710689896870347, 25.00913256532058415028349540196, 26.26407048904031119541032896702, 27.34728685226188592330143516621, 27.930257074485883916326100478097, 30.00622314422540471059637468851, 30.64361017458379875080787967044, 31.503054242471559662839066690681

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