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Properties

Label 1-1384-1384.1037-r0-0-0
Degree $1$
Conductor $1384$
Sign $1$
Analytic cond. $6.42726$
Root an. cond. $6.42726$
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  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s + 31-s + 33-s − 35-s − 37-s − 39-s + 41-s − 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s − 29-s + 31-s + 33-s − 35-s − 37-s − 39-s + 41-s − 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1384 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1384\)    =    \(2^{3} \cdot 173\)
Sign: $1$
Analytic conductor: \(6.42726\)
Root analytic conductor: \(6.42726\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1384} (1037, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1384,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.644743524\)
\(L(\frac12)\) \(\approx\) \(2.644743524\)
\(L(1)\) \(\approx\) \(1.685636613\)
\(L(1)\) \(\approx\) \(1.685636613\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
173 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
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

−20.711807027574321356277781330, −20.03030134124284499837751261911, −19.38846052073786775845310856081, −18.781991273776949330576571230145, −17.7726494300208095107767852392, −17.073567102901096030423477299713, −16.28522547968069479630096042313, −15.348082632397606075913551745892, −14.66929460125138303160535836623, −13.88133336094838280817337172310, −13.31220037319807488105568043537, −12.652629461014356125439132308058, −11.71268842433406246404590699131, −10.43106762380084096534092395929, −9.73302303144535992071088575233, −9.22538649063899626768395123292, −8.64817910138728047309263251905, −7.14288666928026896933769303553, −6.9218645661012476364750289100, −5.82447503060526640897292017521, −4.78015338997037723988454698068, −3.76710081117449134905573831706, −2.86500850007648125543485030839, −2.19924584008229203900906642486, −1.10160252077443150260591472466, 1.10160252077443150260591472466, 2.19924584008229203900906642486, 2.86500850007648125543485030839, 3.76710081117449134905573831706, 4.78015338997037723988454698068, 5.82447503060526640897292017521, 6.9218645661012476364750289100, 7.14288666928026896933769303553, 8.64817910138728047309263251905, 9.22538649063899626768395123292, 9.73302303144535992071088575233, 10.43106762380084096534092395929, 11.71268842433406246404590699131, 12.652629461014356125439132308058, 13.31220037319807488105568043537, 13.88133336094838280817337172310, 14.66929460125138303160535836623, 15.348082632397606075913551745892, 16.28522547968069479630096042313, 17.073567102901096030423477299713, 17.7726494300208095107767852392, 18.781991273776949330576571230145, 19.38846052073786775845310856081, 20.03030134124284499837751261911, 20.711807027574321356277781330

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