Properties

Label 1-344-344.269-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.498 + 0.866i$
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  + (0.222 + 0.974i)3-s + (−0.623 + 0.781i)5-s + 7-s + (−0.900 + 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.623 + 0.781i)13-s + (−0.900 − 0.433i)15-s + (0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.222 + 0.974i)21-s + (−0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.623 − 0.781i)27-s + (0.222 − 0.974i)29-s + (−0.222 + 0.974i)31-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.623 + 0.781i)5-s + 7-s + (−0.900 + 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.623 + 0.781i)13-s + (−0.900 − 0.433i)15-s + (0.623 + 0.781i)17-s + (0.900 + 0.433i)19-s + (0.222 + 0.974i)21-s + (−0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.623 − 0.781i)27-s + (0.222 − 0.974i)29-s + (−0.222 + 0.974i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6396955904 + 1.105554444i\)
\(L(\frac12)\) \(\approx\) \(0.6396955904 + 1.105554444i\)
\(L(1)\) \(\approx\) \(0.9378237169 + 0.5910249557i\)
\(L(1)\) \(\approx\) \(0.9378237169 + 0.5910249557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.222 + 0.974i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−24.53338430962725181255698828662, −23.999351110332729652104999135848, −22.99549363290935938030521941265, −22.13533498067244944445952573783, −20.50850237878504367131702658258, −20.31137408514436977340797621516, −19.384632442858061197311985115316, −18.29599897752185052101609200828, −17.543524980648080021981648151039, −16.75569757593466556838772699607, −15.50423106681463007586297119104, −14.49553136066601169006757243744, −13.819153435155435308577920296590, −12.477080436040715908185903568418, −12.07140952710440209274745949890, −11.210283629962662085752387037364, −9.59565339180759125118834147762, −8.60350900658969036546730894913, −7.74021494859116844797715525083, −7.12837095154403374697649386309, −5.57834414435739586298124425691, −4.6914548087712004103916623113, −3.322174916600030542860483624907, −1.88408210329298457542484954990, −0.82228923208123039600822109641, 1.82009949984194485017197260627, 3.3184406255971232137756503129, 4.03520694262646299775475556929, 5.10290154262186452125296987152, 6.357623085666952593999642490881, 7.69319660405482686709155558428, 8.42730658803682650892067473516, 9.63592378429386847282176569569, 10.48973024741530880421920206461, 11.57303487068452380154558902927, 11.89736270669240642486919064370, 14.05833775551777016918106562600, 14.33507738277609410329785432400, 15.16968345026872545224334904743, 16.16138005387897563878504967042, 17.018063801483931815245486379049, 18.01030697911771879105278039958, 19.239382534751482847443561178977, 19.767422682878021747489770577483, 20.93204236492330672523925634730, 21.702666312888419927677819792115, 22.32090809243793763248886057651, 23.35421637085826920800126613490, 24.264164690838139507153077855596, 25.26277665243146061789627783332

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