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Properties

Label 2-344-344.299-c0-0-0
Degree $2$
Conductor $344$
Sign $0.00612 + 0.999i$
Analytic cond. $0.171678$
Root an. cond. $0.414340$
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  + (−0.900 + 0.433i)2-s + (−1.12 − 0.541i)3-s + (0.623 − 0.781i)4-s + 1.24·6-s + (−0.222 + 0.974i)8-s + (0.346 + 0.433i)9-s + (−1.12 − 1.40i)11-s + (−1.12 + 0.541i)12-s + (−0.222 − 0.974i)16-s + (−0.277 − 1.21i)17-s + (−0.500 − 0.240i)18-s + (0.777 − 0.974i)19-s + (1.62 + 0.781i)22-s + (0.777 − 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−1.12 − 0.541i)3-s + (0.623 − 0.781i)4-s + 1.24·6-s + (−0.222 + 0.974i)8-s + (0.346 + 0.433i)9-s + (−1.12 − 1.40i)11-s + (−1.12 + 0.541i)12-s + (−0.222 − 0.974i)16-s + (−0.277 − 1.21i)17-s + (−0.500 − 0.240i)18-s + (0.777 − 0.974i)19-s + (1.62 + 0.781i)22-s + (0.777 − 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00612 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00612 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.00612 + 0.999i$
Analytic conductor: \(0.171678\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 344,\ (\ :0),\ 0.00612 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2984771271\)
\(L(\frac12)\) \(\approx\) \(0.2984771271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
19 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.623 + 0.781i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.445 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32864874265142334177389415129, −10.77880074622368561453254696789, −9.657308054928504084985079152439, −8.621701970381994698599663438423, −7.61180628445879131253894007592, −6.77391980567259955646682994623, −5.76798947606259489237411905089, −5.15448168191270438007927587333, −2.74102117807717627115368832369, −0.65104441533925965877777233190, 2.03497982295242738357036736634, 3.82713356824279677566174915286, 5.09244492943143590320286297250, 6.17822838415213605813131760136, 7.41281954521229814022744506634, 8.214624546204804128338088742293, 9.635249877532291172871110953003, 10.21138511075846073098387335780, 10.81994181115797612035766625861, 11.79842546078895772734347568040

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