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Properties

Label 1-68-68.59-r1-0-0
Degree $1$
Conductor $68$
Sign $-0.0758 + 0.997i$
Analytic cond. $7.30761$
Root an. cond. $7.30761$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + i·9-s + (0.707 − 0.707i)11-s − 13-s − i·15-s − i·19-s − 21-s + (0.707 − 0.707i)23-s + i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (0.707 + 0.707i)31-s + 33-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + i·9-s + (0.707 − 0.707i)11-s − 13-s − i·15-s − i·19-s − 21-s + (0.707 − 0.707i)23-s + i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s + (0.707 + 0.707i)31-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-0.0758 + 0.997i$
Analytic conductor: \(7.30761\)
Root analytic conductor: \(7.30761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (1:\ ),\ -0.0758 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375420554 + 1.484075077i\)
\(L(\frac12)\) \(\approx\) \(1.375420554 + 1.484075077i\)
\(L(1)\) \(\approx\) \(1.256090415 + 0.6359321071i\)
\(L(1)\) \(\approx\) \(1.256090415 + 0.6359321071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 - iT \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 - T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−31.47033613886957687940718055272, −30.0909092378939103035123785370, −29.43434434377830510720828456954, −28.33967963491238167295893196924, −26.75916762902659918971649883024, −25.684446059097239833455885115389, −24.884579160707873730420552250568, −23.90162523636020903224662605571, −22.59059651331962171794758690724, −21.13807794826086336848844814286, −19.96430318816078885953063706659, −19.39594215595435981828554570443, −17.65709540743454871139709978078, −16.97200819966680153252143768155, −15.2469245620183666736372486633, −13.89563454151916732481593465333, −13.09529575145832982712117176221, −12.04618718626871017552906110726, −9.87415415348972873001441526980, −9.11980831146767397907079600273, −7.48318922223459632891598397190, −6.40254208383725645295607757477, −4.47228137559608128831954786629, −2.63624748608232934746525112837, −1.02083488754548292896924032512, 2.37256756742569653266620423607, 3.48697338096861935326613763009, 5.39521575908269484448143247991, 6.81122135805097076791392278553, 8.65260097866831970804944390106, 9.65967854510628828270382415798, 10.66076169516044554568175469207, 12.39991810910763624879883827409, 13.97556483052767114884819809679, 14.67593613489662908782718774272, 15.94382558984656757726394706544, 17.13082875343772680063793372188, 18.783541092747893030371187722998, 19.499540325824736956028926158985, 21.03664894812534502546518283788, 21.95379526590951466751660575017, 22.603509790359582634611804707275, 24.7693568044980362538399468450, 25.33296219638541561602098353997, 26.5240446532944299562429898871, 27.271604450055072153983041536958, 28.78117800322678544179295824829, 29.7478723147333469102421064359, 31.0311946027180919871282193496, 32.05989221044637060291104859902

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