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Properties

Label 1-344-344.13-r0-0-0
Degree $1$
Conductor $344$
Sign $0.994 - 0.105i$
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  + (−0.0747 + 0.997i)3-s + (−0.955 − 0.294i)5-s + (−0.5 − 0.866i)7-s + (−0.988 − 0.149i)9-s + (−0.623 + 0.781i)11-s + (0.733 − 0.680i)13-s + (0.365 − 0.930i)15-s + (0.955 − 0.294i)17-s + (0.988 − 0.149i)19-s + (0.900 − 0.433i)21-s + (0.365 + 0.930i)23-s + (0.826 + 0.563i)25-s + (0.222 − 0.974i)27-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)3-s + (−0.955 − 0.294i)5-s + (−0.5 − 0.866i)7-s + (−0.988 − 0.149i)9-s + (−0.623 + 0.781i)11-s + (0.733 − 0.680i)13-s + (0.365 − 0.930i)15-s + (0.955 − 0.294i)17-s + (0.988 − 0.149i)19-s + (0.900 − 0.433i)21-s + (0.365 + 0.930i)23-s + (0.826 + 0.563i)25-s + (0.222 − 0.974i)27-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.994 - 0.105i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9028152687 - 0.04772200580i\)
\(L(\frac12)\) \(\approx\) \(0.9028152687 - 0.04772200580i\)
\(L(1)\) \(\approx\) \(0.8453555918 + 0.08262997915i\)
\(L(1)\) \(\approx\) \(0.8453555918 + 0.08262997915i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.988 - 0.149i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.0747 + 0.997i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−24.78748097949097189904951883087, −23.94276648909551511521282065037, −23.28072702847014839177926245359, −22.49728160812518051618079510597, −21.45055416137921215121504914861, −20.277922912256178341259753691828, −19.29920841390879960715741066700, −18.50872447302574776558608987618, −18.42066654452165630571025099512, −16.64884756498803738797259321773, −16.07634094366156912541715304390, −14.96390814004455316130807337588, −13.980673721187586479271830366816, −13.00449523451524384386586231976, −12.04768302839450385772143033000, −11.52884638131758144949462728194, −10.37647321697237168546890759990, −8.75045465287517080354915634443, −8.227388269580269873843844672598, −7.090825104118076196144827727834, −6.21930583212295822763321934055, −5.20660212100427041909012010033, −3.4449928740262280794245279575, −2.7339984395787470082701796755, −1.08240446294105411257054170089, 0.73091774063456234151651434720, 3.04565581458488671785652978996, 3.78738646847561360152726400445, 4.76037758379573606116232502482, 5.76882672572450063604463333144, 7.367431187832027286627528141153, 8.02030741280527093936982522939, 9.38773255657010110838372267492, 10.12745614969257480927765801277, 11.05431809056306673918008725002, 11.958197260548429924503138038942, 13.08830256743972360338651578604, 14.093901769723901622977073351590, 15.46491273048808209947292839299, 15.66338624211367815458770844636, 16.67215020166825835496963522390, 17.49401655513958533173986385440, 18.771686862873145477011020637800, 19.887567168128689228371891693635, 20.44844433894255551331617731987, 21.08891304713387620273041497735, 22.52682556150963353296805192521, 23.071966496384627023995781180993, 23.56855469451026631632881817706, 25.036405861013033053997802224690

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