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Properties

Label 2-344-344.307-c0-0-2
Degree $2$
Conductor $344$
Sign $-0.675 + 0.736i$
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  − i·2-s + (0.5 − 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (−0.866 + 0.499i)15-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)20-s + ⋯
L(s)  = 1  − i·2-s + (0.5 − 0.866i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (−0.866 + 0.499i)15-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.675 + 0.736i$
Analytic conductor: \(0.171678\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 344,\ (\ :0),\ -0.675 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7894340620\)
\(L(\frac12)\) \(\approx\) \(0.7894340620\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
43 \( 1 - T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.66816722670820874722035010247, −10.66665199776799329327932811956, −9.589259141013978397311475751477, −8.366337567145744942705266362830, −7.984945030205546112146272127457, −6.94666805328297755627967678859, −4.90859190639520123978047427718, −4.30349067174448724227589066457, −2.70717422503861292880387509613, −1.38417145639177018939575288884, 3.02678504228193590009823131591, 4.32057309900276481865479764108, 4.93071450025309771744318422281, 6.43182498802814415082427115615, 7.45742815147500585503932639410, 8.380419719067662273930664726490, 8.978534177651026665859593963948, 10.10283309793632065777630043295, 10.96647478672468670330855690544, 12.12085497418785517721645930148

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