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Properties

Label 2-344-344.251-c0-0-1
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: $\chi_{344} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 344,\ (\ :0),\ 0.675 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8934605608\)
\(L(\frac12)\) \(\approx\) \(0.8934605608\)
\(L(1)\) 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

−11.43892218134403626694784489091, −10.43002851269860460367873957157, −9.861504055201280719714889437540, −9.098324020542616774322217796188, −8.536404654028930060931233300791, −6.66985060608640742187138558196, −5.44065051016998673062119244974, −4.12375753650335690679204930259, −3.50329066391045781060409644117, −1.86415586974603818499124393960, 2.16208646345264748183519634425, 3.62315986760329884325173439544, 5.43236127828263893255235513454, 6.28135879433564158875403452688, 6.94072992364929510907279789219, 8.008897018636522946122642055597, 8.861197996240794868574592850597, 9.769290883412713104815474027118, 10.66018339113145026417090850381, 12.38102454201630955205571035486

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