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Properties

Label 1-344-344.171-r0-0-0
Degree $1$
Conductor $344$
Sign $1$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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 + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s + 57-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 + 45-s − 47-s + 49-s − 51-s − 53-s + 55-s + 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $1$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{344} (171, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283586129\)
\(L(\frac12)\) \(\approx\) \(1.283586129\)
\(L(1)\) \(\approx\) \(1.072201635\)
\(L(1)\) \(\approx\) \(1.072201635\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
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 \( 1 + T \)
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

−24.76246894561095659048512374147, −24.0496364870473633209533568949, −23.14837580757205032694636089503, −21.95939140457325105153426554355, −21.67890070258393173654999851079, −20.73518854437325263247508152930, −19.490230846253459517629892385118, −18.363970356374026111461285514048, −17.56220029084775654294587989970, −17.08783140179098459202014797862, −16.24848165174922783031665483863, −14.67111121818761374323386112278, −14.30585033239057410235004564871, −12.88109831234555227028850773744, −12.090641043303890726164992141087, −11.19438489392983535442566364290, −10.19711050489384935373995420686, −9.45797630622248095890347078416, −8.0675173923306695263549477569, −6.87296120520845005395232987807, −5.958841709286804013374399375057, −5.08635384009794552134170168694, −4.146012962327244471424799516067, −2.210455185276736988739479147059, −1.218462406742651288412854104739, 1.218462406742651288412854104739, 2.210455185276736988739479147059, 4.146012962327244471424799516067, 5.08635384009794552134170168694, 5.958841709286804013374399375057, 6.87296120520845005395232987807, 8.0675173923306695263549477569, 9.45797630622248095890347078416, 10.19711050489384935373995420686, 11.19438489392983535442566364290, 12.090641043303890726164992141087, 12.88109831234555227028850773744, 14.30585033239057410235004564871, 14.67111121818761374323386112278, 16.24848165174922783031665483863, 17.08783140179098459202014797862, 17.56220029084775654294587989970, 18.363970356374026111461285514048, 19.490230846253459517629892385118, 20.73518854437325263247508152930, 21.67890070258393173654999851079, 21.95939140457325105153426554355, 23.14837580757205032694636089503, 24.0496364870473633209533568949, 24.76246894561095659048512374147

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