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Properties

Label 1-305-305.243-r1-0-0
Degree $1$
Conductor $305$
Sign $0.850 - 0.525i$
Analytic cond. $32.7768$
Root an. cond. $32.7768$
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  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − 11-s − i·12-s + i·13-s − 14-s + 16-s + i·17-s − i·18-s − 19-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − 11-s − i·12-s + i·13-s − 14-s + 16-s + i·17-s − i·18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 305 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(305\)    =    \(5 \cdot 61\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(32.7768\)
Root analytic conductor: \(32.7768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{305} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 305,\ (1:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2508739629 + 0.07126803551i\)
\(L(\frac12)\) \(\approx\) \(-0.2508739629 + 0.07126803551i\)
\(L(1)\) \(\approx\) \(0.3782889217 + 0.6120843330i\)
\(L(1)\) \(\approx\) \(0.3782889217 + 0.6120843330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
61 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 - iT \)
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

−23.92969305544868970055757352422, −23.24207673816330680539021620219, −22.74498130299926304328097019873, −21.36398132301664815686856507166, −20.47402677200106727521195057579, −19.78395268868133420677220908510, −19.017686036589472872418697425065, −17.84192089019378567368929608957, −17.64254940953037602370176171889, −16.25410741308730948858899166045, −14.691919785445769379019552552173, −13.70898528592708152786417865098, −13.0891000200847953922363387623, −12.37245673023316852620387971668, −11.115045799098996998468088858191, −10.546600296173334770691584380874, −9.3021460165744296540779402089, −8.01409069890637758854724989871, −7.42168216044179468581796958801, −5.86290063266501322143183374007, −4.75624051294149088786344829000, −3.32035203303010820896120832441, −2.382834946875695344655796998013, −1.02917907859851450752692111560, −0.09029431541262276628645005712, 2.39648274075954955867456004418, 3.88258404783463717893221784523, 4.83905060200408314952365652956, 5.73086027647352424982592701591, 6.666884364065823770027916173419, 8.34009808047038041287660347802, 8.69891608946843446290951819791, 9.85462732364511930394123482419, 10.75204219034263757592791511502, 12.15189061332745383221591472584, 13.18309084125457734335714159350, 14.484311484235684443898636789507, 15.00834504720961424788336911710, 15.89308998663079033462402966222, 16.550197945583567777113863301177, 17.52202647389983465703245603284, 18.56124145054602312755757191976, 19.36228888605311807043339997927, 20.926260316255417836682305692201, 21.60519313944342260774529956592, 22.23618112110002296897464178732, 23.387540586028269959221253896328, 24.01572238674012234469443146773, 25.26240111778649551761995997233, 25.91550572883776474123639328212

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