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Properties

Label 1-847-847.145-r0-0-0
Degree $1$
Conductor $847$
Sign $0.323 + 0.946i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.999 + 0.0380i)2-s + (−0.669 − 0.743i)3-s + (0.997 − 0.0760i)4-s + (−0.749 + 0.662i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (−0.104 + 0.994i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (0.988 − 0.151i)16-s + (−0.683 + 0.730i)17-s + (0.0665 − 0.997i)18-s + (0.483 − 0.875i)19-s + (−0.696 + 0.717i)20-s + ⋯
L(s)  = 1  + (−0.999 + 0.0380i)2-s + (−0.669 − 0.743i)3-s + (0.997 − 0.0760i)4-s + (−0.749 + 0.662i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (−0.104 + 0.994i)9-s + (0.723 − 0.690i)10-s + (−0.723 − 0.690i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (0.988 − 0.151i)16-s + (−0.683 + 0.730i)17-s + (0.0665 − 0.997i)18-s + (0.483 − 0.875i)19-s + (−0.696 + 0.717i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.323 + 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3520174651 + 0.2517152132i\)
\(L(\frac12)\) \(\approx\) \(0.3520174651 + 0.2517152132i\)
\(L(1)\) \(\approx\) \(0.4817194290 + 0.01512358961i\)
\(L(1)\) \(\approx\) \(0.4817194290 + 0.01512358961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 + 0.0380i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (-0.749 + 0.662i)T \)
13 \( 1 + (0.941 + 0.336i)T \)
17 \( 1 + (-0.683 + 0.730i)T \)
19 \( 1 + (0.483 - 0.875i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (0.991 - 0.132i)T \)
37 \( 1 + (-0.851 + 0.524i)T \)
41 \( 1 + (-0.985 - 0.170i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.0665 + 0.997i)T \)
53 \( 1 + (0.988 + 0.151i)T \)
59 \( 1 + (-0.345 + 0.938i)T \)
61 \( 1 + (-0.532 + 0.846i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.935 + 0.353i)T \)
79 \( 1 + (-0.861 - 0.508i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (-0.198 + 0.980i)T \)
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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

−21.74165449962174990749186602457, −20.921551339593664483487180446432, −20.33580699000479074221118382652, −19.70968488600311245173186624983, −18.57016274612555476874612820288, −17.92307238243634430013943246524, −17.10453984827849011997893280692, −16.30290930323898794624047331911, −15.73952716695635194489091703778, −15.35824702911332339212265994898, −13.95593990708170033363088070799, −12.62256851839139818021184977410, −11.750855183415406383934624657951, −11.40832719235727015120521864797, −10.33212802298143904965636056256, −9.705032863741448683985805028880, −8.67267349357120933474184922456, −8.150970606344852859485184592806, −6.97678678208144384228246202837, −6.04151570267988674094786532176, −5.10871211300624536472105395421, −3.98159064295066059967894104452, −3.15365975815331908645617336185, −1.4879626948146792162453487449, −0.38545615727697073390201444608, 0.97613401197997556533877455730, 2.10100149838822086678366497430, 3.12934879029103537503715390578, 4.45855754743803715779207565143, 5.914645120932215780027606782, 6.633841500047194896918285231666, 7.178335686004792006769784943491, 8.2278353027072785649533094135, 8.72322499384322827719651922326, 10.316109174116284333192998546690, 10.70232184976311259790276156478, 11.724951015848303474603052980675, 11.98117472295006662916700357596, 13.28428135459862267791265605267, 14.21011861018128264243080619188, 15.49825321820593649767947871415, 15.84646964035100521624459699013, 16.85139360306830941752471184213, 17.68505658000086129065233978170, 18.282241312560427089447438907091, 18.922551456107282198998575533640, 19.61880181930754033745373861401, 20.27642027404051283043043187140, 21.53630061268524126399027727828, 22.32880971715402532842800839894

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