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Properties

Label 1-105-105.104-r0-0-0
Degree $1$
Conductor $105$
Sign $1$
Analytic cond. $0.487617$
Root an. cond. $0.487617$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 11-s + 13-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 29-s − 31-s + 32-s − 34-s − 37-s − 38-s + 41-s − 43-s − 44-s + 46-s − 47-s + 52-s + 53-s − 58-s + 59-s − 61-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 11-s + 13-s + 16-s − 17-s − 19-s − 22-s + 23-s + 26-s − 29-s − 31-s + 32-s − 34-s − 37-s − 38-s + 41-s − 43-s − 44-s + 46-s − 47-s + 52-s + 53-s − 58-s + 59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.487617\)
Root analytic conductor: \(0.487617\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 105,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.777644139\)
\(L(\frac12)\) \(\approx\) \(1.777644139\)
\(L(1)\) \(\approx\) \(1.720148986\)
\(L(1)\) \(\approx\) \(1.720148986\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
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

−29.81183932642223432305890548289, −28.91694642959154523629622140391, −27.93650268603585835006525084610, −26.32200911279995332683334128787, −25.509932190826191270454116721049, −24.31915969183197091773299882641, −23.448412084519404177850939942, −22.589851476453794840498219310699, −21.34257419248581940898644944004, −20.68405044142393211855232045771, −19.47034002879828613436007948109, −18.20461932859221706711004561700, −16.73636122039523014164316730611, −15.67012050348698231277615318673, −14.827120768810874534393445019306, −13.39081440029833406196735481678, −12.862345237901543307432137965867, −11.29907981354199366805953260575, −10.555559845234520329443114663035, −8.68713481050641910740736755603, −7.26120229959507451601179000344, −6.04532098532612637992106866437, −4.83316627385172929886241582021, −3.49996851859653038990910140652, −2.04884908157592245734866028139, 2.04884908157592245734866028139, 3.49996851859653038990910140652, 4.83316627385172929886241582021, 6.04532098532612637992106866437, 7.26120229959507451601179000344, 8.68713481050641910740736755603, 10.555559845234520329443114663035, 11.29907981354199366805953260575, 12.862345237901543307432137965867, 13.39081440029833406196735481678, 14.827120768810874534393445019306, 15.67012050348698231277615318673, 16.73636122039523014164316730611, 18.20461932859221706711004561700, 19.47034002879828613436007948109, 20.68405044142393211855232045771, 21.34257419248581940898644944004, 22.589851476453794840498219310699, 23.448412084519404177850939942, 24.31915969183197091773299882641, 25.509932190826191270454116721049, 26.32200911279995332683334128787, 27.93650268603585835006525084610, 28.91694642959154523629622140391, 29.81183932642223432305890548289

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