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Properties

Label 1-2352-2352.83-r1-0-0
Degree $1$
Conductor $2352$
Sign $0.890 - 0.455i$
Analytic cond. $252.757$
Root an. cond. $252.757$
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  + (−0.974 + 0.222i)5-s + (−0.433 + 0.900i)11-s + (−0.433 + 0.900i)13-s + (0.623 − 0.781i)17-s − i·19-s + (−0.623 − 0.781i)23-s + (0.900 − 0.433i)25-s + (−0.781 − 0.623i)29-s + 31-s + (−0.781 − 0.623i)37-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + (0.900 + 0.433i)47-s + (−0.781 + 0.623i)53-s + (0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)5-s + (−0.433 + 0.900i)11-s + (−0.433 + 0.900i)13-s + (0.623 − 0.781i)17-s − i·19-s + (−0.623 − 0.781i)23-s + (0.900 − 0.433i)25-s + (−0.781 − 0.623i)29-s + 31-s + (−0.781 − 0.623i)37-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + (0.900 + 0.433i)47-s + (−0.781 + 0.623i)53-s + (0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(252.757\)
Root analytic conductor: \(252.757\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2352,\ (1:\ ),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7661671299 - 0.1845899968i\)
\(L(\frac12)\) \(\approx\) \(0.7661671299 - 0.1845899968i\)
\(L(1)\) \(\approx\) \(0.7451349879 + 0.09250783565i\)
\(L(1)\) \(\approx\) \(0.7451349879 + 0.09250783565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.433 + 0.900i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.781 - 0.623i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.781 + 0.623i)T \)
59 \( 1 + (-0.974 - 0.222i)T \)
61 \( 1 + (-0.781 - 0.623i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (0.433 + 0.900i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−19.28353730009873527440215901870, −19.057928817170725090067406392955, −18.02960823588487404840169780885, −17.26461725153974906991916032149, −16.58674509328066584331465154030, −15.707724971637188312998873317751, −15.37002257888709799750687497157, −14.546878718811706695942619041005, −13.57617258610207836642760936418, −12.96125234449849619816563117236, −12.1458771753419320374764411801, −11.54848516968957198902379118132, −10.703779745756816649979548240511, −10.13516882383736444364861610430, −8.98771491265090810330392422524, −8.337693383937743506797761398421, −7.71344673836334954515742659830, −7.01681870543799556309895187734, −5.8579964170771895908159642839, −5.24952646136781371966797342161, −4.33570284230668941988708550885, −3.38439324803087547914114977900, −2.90914879222297046944506404859, −1.51663695470504099570810245679, −0.498820809625490975079748383892, 0.24674656523805786513181525640, 1.57271747790351422506455239967, 2.50731768810490893138669589095, 3.40617459306626994161581104731, 4.35548719927894696278681839930, 4.80012455613003767031877448759, 5.98161591864430409711573736967, 6.84315167533152152718086582603, 7.626527136109998817035325927853, 8.016944812874953795834673041569, 9.13024155380570976507987579227, 9.88451834274290048775726102546, 10.56526469099688941069341648181, 11.50696537243154759496850476513, 12.17374533066505030253006000454, 12.50829464627594071089372734944, 13.759263548373869808595705339820, 14.37545106350199632678294861558, 15.07040191042537770517579637775, 15.76560062723643155504016262896, 16.456646782617111934654393952469, 17.077272656391085067915809167932, 18.127954087124914291091792574167, 18.73519179928141229272447084648, 19.1896035167213389648977122562

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