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Properties

Label 1-351-351.2-r0-0-0
Degree $1$
Conductor $351$
Sign $0.0838 + 0.996i$
Analytic cond. $1.63003$
Root an. cond. $1.63003$
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  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.984 + 0.173i)5-s + (0.342 − 0.939i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + (−0.342 + 0.939i)11-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + 17-s + (0.866 + 0.5i)19-s + (−0.342 + 0.939i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.984 + 0.173i)5-s + (0.342 − 0.939i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + (−0.342 + 0.939i)11-s + (0.939 − 0.342i)14-s + (−0.939 − 0.342i)16-s + 17-s + (0.866 + 0.5i)19-s + (−0.342 + 0.939i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (0.939 + 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.0838 + 0.996i$
Analytic conductor: \(1.63003\)
Root analytic conductor: \(1.63003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 351,\ (0:\ ),\ 0.0838 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.535198944 + 1.411415943i\)
\(L(\frac12)\) \(\approx\) \(1.535198944 + 1.411415943i\)
\(L(1)\) \(\approx\) \(1.449344764 + 0.8031456791i\)
\(L(1)\) \(\approx\) \(1.449344764 + 0.8031456791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (0.342 - 0.939i)T \)
11 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.642 - 0.766i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.642 - 0.766i)T \)
53 \( 1 - T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−24.46422377782196594440232956800, −23.818826701559664803300062497247, −22.49599715495161370938292112707, −21.858629579457533464976768665092, −21.17384964528082159477227706800, −20.55448616525262428023012389166, −19.30875648267040395342951939115, −18.46446943592047574083809030579, −17.85846120017065859406800082035, −16.47113529321275447180050782592, −15.472792611885985935508255305301, −14.35226466293024483899433165754, −13.78598848154722148319151123514, −12.80716880755490494007083372104, −11.932687046833853769801562257934, −11.04290374705914861622262441727, −9.9503527074895314469904056438, −9.22714219489650607650264424211, −8.10082105473141774392546067385, −6.253783692883445545765980865291, −5.62561601058806290972385540452, −4.81763529985092199884962424840, −3.234823452918193927357829566294, −2.38455335743607000306235295433, −1.21361715370721538580971570966, 1.67423879862402424517818055795, 3.088367131954161277692022148767, 4.28225927702388383848369888182, 5.29390618550401755282550704557, 6.16469995122779028051532764375, 7.34629947202367728910167741713, 7.89187487910046337777675045681, 9.47606602598912873803758303990, 10.18788723393100404866755452367, 11.53282846358144563201364384219, 12.66782612696465551446288739065, 13.47308359936774887983112012773, 14.281468524760121931480323155272, 14.867162150902099729419107608804, 16.20819860511726938034315861054, 16.89196287671557338387692903728, 17.79163227949319308731932089587, 18.34567927570072057693684919104, 20.11695549407426942580712677847, 20.81555054213087036510107237175, 21.56683808559438742970568401999, 22.63004255182549917310475980329, 23.20044107906603808128148543072, 24.18302030333983118485788661934, 24.976153576713177637044605214770

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