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Properties

Label 1-1309-1309.769-r0-0-0
Degree $1$
Conductor $1309$
Sign $-0.615 + 0.788i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
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  + 2-s + i·3-s + 4-s + i·5-s + i·6-s + 8-s − 9-s + i·10-s + i·12-s + 13-s − 15-s + 16-s − 18-s − 19-s + i·20-s + ⋯
L(s)  = 1  + 2-s + i·3-s + 4-s + i·5-s + i·6-s + 8-s − 9-s + i·10-s + i·12-s + 13-s − 15-s + 16-s − 18-s − 19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ -0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273580033 + 2.610178306i\)
\(L(\frac12)\) \(\approx\) \(1.273580033 + 2.610178306i\)
\(L(1)\) \(\approx\) \(1.604906874 + 1.076956250i\)
\(L(1)\) \(\approx\) \(1.604906874 + 1.076956250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 + iT \)
19 \( 1 \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + iT \)
37 \( 1 \)
41 \( 1 + iT \)
43 \( 1 + T \)
47 \( 1 \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + iT \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 + iT \)
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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

−20.66699950864997337357434312436, −20.18830319640649810134264357345, −19.21364631283110695373408362872, −18.67799892935062483278735066584, −17.38099380307933626740971440917, −16.93456422916555391114105039025, −16.039263975146052323687246083813, −15.279060679436874237883174629705, −14.271645588117752205961707846461, −13.627733294064778730788951010746, −12.8366864129406192223486435259, −12.59964731531473803301685546819, −11.56198115629279229077272944571, −11.04261690845566296610761465552, −9.78923469727186346260940747483, −8.43680851332168523178791182218, −8.17441268869377324452715765386, −6.98112687519108333798128347045, −6.19376324181023346520236884612, −5.62475099254194605868217702975, −4.55692771910796550189419391893, −3.799450457976446926679147411436, −2.552709284380171601663577106099, −1.78514596101158623231393728180, −0.772285405936870733664670945941, 1.698581841856292609983373891942, 2.83793440874370724689949551988, 3.48787049254209302698190734281, 4.14993435012424289167487264753, 5.16249912413891135766674850762, 5.9997233305187962632062833058, 6.63892404985798793712360541192, 7.67679345770861800328719965177, 8.686749398477298376287596880656, 9.77465180160820919785282818887, 10.74147319615823360050734625302, 10.96225902354748829291986727849, 11.799545951561620752558214277664, 12.88039793195489668053546410348, 13.75730148363822258846827597225, 14.4895897202538644016588262355, 14.94058399955348290980256872066, 15.845078583615020272842833959249, 16.19781403574308178217896396696, 17.31382944788065393395737507935, 18.10388370199799909030828857598, 19.32991023073393270912351762081, 19.779226348293343919213139158358, 20.8772850181378611182411478289, 21.34079032687796952254341739754

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