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Properties

Label 1-344-344.221-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.736 + 0.675i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s − 21-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.736 + 0.675i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.736 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4724719948 + 1.214028243i\)
\(L(\frac12)\) \(\approx\) \(0.4724719948 + 1.214028243i\)
\(L(1)\) \(\approx\) \(0.9369059797 + 0.6705109658i\)
\(L(1)\) \(\approx\) \(0.9369059797 + 0.6705109658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−24.45369824303406764886170121650, −23.74625527873455931602423875724, −23.20292770921647934213232226888, −21.74390913109802738438125582401, −20.797051325997396799426115234662, −20.07331928266550500708664056056, −19.433044722923755435211404875093, −18.12765155700157752115200183533, −17.69781624244568903946903075418, −16.3910459270007539761128552652, −15.81793831059240814618905727990, −14.17071494719546867022006602275, −13.55401158938094592886755256542, −13.0301154994109157357499308082, −12.000794317643523650920833717618, −10.80265143750699271764860658139, −9.46775633658095239165670003836, −8.89075301131418460258709432985, −7.637047570086555518400099252715, −6.89543527725252408139191742145, −5.73269858485985530581160726214, −4.51580771504845409455711346490, −3.169278597639401274733777760495, −1.960084799851382586001509037360, −0.74061279184580431034758369830, 2.29521336173478233662698948799, 2.917206789511159497691826926233, 4.05032315156863491331920680197, 5.58019857274307198580325586109, 6.10012089304188723709668188951, 7.76454970774081321020616381454, 8.57629039928945591448494883811, 9.79883587076247536743624552172, 10.32787956368440928525878017033, 11.2228542364835974447753172392, 12.68571575269762890371835693786, 13.55909876585950825503611378539, 14.59266078780145770549159362603, 15.39143171750078332804933675001, 15.91122068365262492276495687763, 17.16837904594372916489034553600, 18.349320445484521239854318638821, 18.83946700932029736447834186825, 20.07588891956340571067622661049, 20.91182678109918881583764142447, 21.75940690051488248945683624072, 22.37786196274548321000493938482, 23.14523388988132648344040977350, 24.67961366348829058009562304674, 25.41988874686210395363507820223

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