Properties

Label 1-5-5.4-r0-0-0
Degree $1$
Conductor $5$
Sign $1$
Analytic cond. $0.0232199$
Root an. cond. $0.0232199$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

After the Riemann zeta function, the analytic conductor of this L-function is the smallest among L-functions of degree 1.

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5\)
Sign: $1$
Analytic conductor: \(0.0232199\)
Root analytic conductor: \(0.0232199\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 5,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2317509475\)
\(L(\frac12)\) \(\approx\) \(0.2317509475\)
\(L(1)\) \(\approx\) \(0.4304089409\)
\(L(1)\) \(\approx\) \(0.4304089409\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
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

−55.58928033540481015096458899849, −53.83044519544216335105426902233, −52.1259022313169741886041306959, −51.08775192674649135525825720413, −48.34566182106784617654820130449, −46.49272715949140534533919935167, −45.4273000827822893888415619096, −44.03129006144169504470090805842, −41.84243854579169430850930688531, −39.56057294640318170505509505995, −38.12918472143653185015141827037, −35.86863837181227459459504863887, −34.728812978904808674143729833981, −33.00045600687051436794975917721, −29.70790935048096556923098651865, −28.46103510017752247518697827232, −26.77609594800414011652357496527, −24.58846621740819520765626997608, −22.227405454459410911877624963081, −19.54073262278475025037869002299, −17.566994292325555202701595268144, −16.03382112838423567459325378224, −11.95884562608351453026565868826, −9.831444432886669616348321347458, −6.64845334472771471612327845997, 6.64845334472771471612327845997, 9.831444432886669616348321347458, 11.95884562608351453026565868826, 16.03382112838423567459325378224, 17.566994292325555202701595268144, 19.54073262278475025037869002299, 22.227405454459410911877624963081, 24.58846621740819520765626997608, 26.77609594800414011652357496527, 28.46103510017752247518697827232, 29.70790935048096556923098651865, 33.00045600687051436794975917721, 34.728812978904808674143729833981, 35.86863837181227459459504863887, 38.12918472143653185015141827037, 39.56057294640318170505509505995, 41.84243854579169430850930688531, 44.03129006144169504470090805842, 45.4273000827822893888415619096, 46.49272715949140534533919935167, 48.34566182106784617654820130449, 51.08775192674649135525825720413, 52.1259022313169741886041306959, 53.83044519544216335105426902233, 55.58928033540481015096458899849

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