Properties

Label 1-41-41.15-r1-0-0
Degree $1$
Conductor $41$
Sign $0.962 + 0.270i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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  + (0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.891 − 0.453i)6-s + (0.891 + 0.453i)7-s + (0.587 + 0.809i)8-s i·9-s + (−0.809 + 0.587i)10-s + (0.156 − 0.987i)11-s + (0.987 − 0.156i)12-s + (−0.453 − 0.891i)13-s + (0.707 + 0.707i)14-s + (0.156 + 0.987i)15-s + (0.309 + 0.951i)16-s + (−0.987 − 0.156i)17-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.891 − 0.453i)6-s + (0.891 + 0.453i)7-s + (0.587 + 0.809i)8-s i·9-s + (−0.809 + 0.587i)10-s + (0.156 − 0.987i)11-s + (0.987 − 0.156i)12-s + (−0.453 − 0.891i)13-s + (0.707 + 0.707i)14-s + (0.156 + 0.987i)15-s + (0.309 + 0.951i)16-s + (−0.987 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ 0.962 + 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.927373363 + 0.4032445846i\)
\(L(\frac12)\) \(\approx\) \(2.927373363 + 0.4032445846i\)
\(L(1)\) \(\approx\) \(2.104027268 + 0.2082876971i\)
\(L(1)\) \(\approx\) \(2.104027268 + 0.2082876971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (0.891 + 0.453i)T \)
11 \( 1 + (0.156 - 0.987i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (-0.987 - 0.156i)T \)
19 \( 1 + (-0.453 + 0.891i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.987 - 0.156i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.891 + 0.453i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−34.057406252223451788926743556541, −33.1189203353543178192161318018, −32.1581399752007298570653522503, −31.07849561686181681352470507448, −30.49126293052938243079234783759, −28.5695043391108741066398760893, −27.67985093306591755699899461099, −26.2743400841518926730974664583, −24.66021924370373337973376271310, −23.88568752132014274651849435950, −22.3878169997118998776124429520, −21.076831387115838512502583874076, −20.316424153966181041002026927306, −19.44595876408721134865614997742, −16.98819538978543739002615856652, −15.59846819740329598525132726017, −14.683221841526731415370067333659, −13.43739443363329421394436981697, −11.95340058196998665049844836823, −10.62998788634676038963245831605, −8.96266504287673162023166603896, −7.29436541067047397848501848483, −4.6920076043228126341351469018, −4.27448147541150950147338892159, −2.0596443432589604164575716242, 2.36490278380049334141559350629, 3.753301945376491067748609998865, 5.88447004614549439130345827281, 7.40553143818840763390146956124, 8.35117930629260651266179586654, 11.07945140925830638125061391896, 12.18585272548723553661009229181, 13.66980510159571693117356199282, 14.69280022726624804819448394012, 15.52507704188104891139350123603, 17.584835359070717908431176866153, 18.967481037266530124871238388986, 20.19834951926494660400617898427, 21.52963480252251286107592737702, 22.79477740692756125506512047952, 24.103175606848655435122298847862, 24.772505498930938967935757148341, 26.10737445868220921270251305119, 27.26917095488785433653471280437, 29.547342888474785027065517147564, 30.178349726155191225846251735222, 31.32719697976162095543389141098, 31.86257315558216568197616877782, 33.59913951115444738863967937415, 34.66425176352297951288214374810

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