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Properties

Label 1-4235-4235.1154-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.379 + 0.925i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.841 + 0.540i)2-s + 3-s + (0.415 + 0.909i)4-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + 9-s + (0.415 + 0.909i)12-s + (−0.415 − 0.909i)13-s + (−0.654 + 0.755i)16-s + (0.959 − 0.281i)17-s + (0.841 + 0.540i)18-s + (−0.959 − 0.281i)19-s + (0.654 − 0.755i)23-s + (−0.142 + 0.989i)24-s + (0.142 − 0.989i)26-s + 27-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)2-s + 3-s + (0.415 + 0.909i)4-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + 9-s + (0.415 + 0.909i)12-s + (−0.415 − 0.909i)13-s + (−0.654 + 0.755i)16-s + (0.959 − 0.281i)17-s + (0.841 + 0.540i)18-s + (−0.959 − 0.281i)19-s + (0.654 − 0.755i)23-s + (−0.142 + 0.989i)24-s + (0.142 − 0.989i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.379 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.914080770 + 2.624502303i\)
\(L(\frac12)\) \(\approx\) \(3.914080770 + 2.624502303i\)
\(L(1)\) \(\approx\) \(2.283698537 + 0.9197269366i\)
\(L(1)\) \(\approx\) \(2.283698537 + 0.9197269366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (0.959 - 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−18.7930443491526506206152890882, −17.63279566627897816313504387617, −16.73528111630518235678206386815, −15.95559847861343533768453302177, −15.310316434662466318792407173018, −14.55062189352723307528873659681, −14.303282750394388361009810942957, −13.50031676178142097354814124769, −12.88568323437152316641083943826, −12.24036638967664043007285569738, −11.602879625453723815574621430145, −10.6002291446035636994135959817, −10.125371148971463005898120453252, −9.28859439380858718409652704976, −8.78932136018982802844903340258, −7.654878384031905810363065916089, −7.14132251860457519676577831697, −6.25538744214847436015874284933, −5.455303279290076456945089657714, −4.51973820914009940378479713890, −3.94159502832424060277754732886, −3.31700734308461707759355314656, −2.34621997575526899995377864905, −1.923088698573928562547865342444, −0.91434993133059340544409757961, 1.081970194431012668029368823847, 2.24963554107279971892085293993, 2.961552441570995835357588836813, 3.3774818788148049032964679913, 4.48425240001604165321866253129, 4.88144380655217725575267458976, 5.8651911493964989137690157914, 6.72759590231264008072197211865, 7.317325649846358365696605099755, 8.07383379283430430979673211050, 8.549736093931363747089930812777, 9.357666047220072062220664759567, 10.35301354092906840928104111484, 10.86303152995686958875992879021, 12.21672969424992935878263647446, 12.410702155290308442624720935155, 13.23719066275634036997025676180, 13.79133468911910390325511018075, 14.6247424696869927510181738124, 14.882564264694892216383322351235, 15.58561448492563857167674636394, 16.3119020029347806099526959263, 16.93548122343944443912200901692, 17.787584387696351980651505023350, 18.456833986521609963131983568943

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