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Properties

Label 1-5-5.4-r0-0-0
Degree 11
Conductor 55
Sign 11
Analytic cond. 0.02321990.0232199
Root an. cond. 0.02321990.0232199
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

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

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

Invariants

Degree: 11
Conductor: 55
Sign: 11
Analytic conductor: 0.02321990.0232199
Root analytic conductor: 0.02321990.0232199
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ5(4,⋅)\chi_{5} (4, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 5, (0: ), 1)(1,\ 5,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.23175094750.2317509475
L(12)L(\frac12) ≈\approx 0.23175094750.2317509475
L(1)L(1) ≈\approx 0.43040894090.4304089409
L(1)L(1) ≈\approx 0.43040894090.4304089409

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
good2 1−T 1 - T
3 1−T 1 - T
7 1−T 1 - T
11 1+T 1 + T
13 1−T 1 - T
17 1−T 1 - T
19 1+T 1 + T
23 1−T 1 - T
29 1+T 1 + T
31 1+T 1 + T
37 1−T 1 - T
41 1+T 1 + T
43 1−T 1 - T
47 1−T 1 - T
53 1−T 1 - T
59 1+T 1 + T
61 1+T 1 + T
67 1−T 1 - T
71 1+T 1 + T
73 1−T 1 - T
79 1+T 1 + T
83 1−T 1 - T
89 1+T 1 + T
97 1−T 1 - T
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   L(s)=∏p (1−αp p−s)−1L(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 ZZ-function along the critical line