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Properties

Label 2-234-117.103-c1-0-12
Degree $2$
Conductor $234$
Sign $-0.0806 + 0.996i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (1.21 − 1.23i)3-s + (0.499 − 0.866i)4-s + (−2.73 − 1.57i)5-s + (0.434 − 1.67i)6-s + (−1.36 + 0.787i)7-s − 0.999i·8-s + (−0.0487 − 2.99i)9-s − 3.15·10-s + (3.26 − 1.88i)11-s + (−0.461 − 1.66i)12-s + (0.899 + 3.49i)13-s + (−0.787 + 1.36i)14-s + (−5.27 + 1.45i)15-s + (−0.5 − 0.866i)16-s + 7.06·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.701 − 0.712i)3-s + (0.249 − 0.433i)4-s + (−1.22 − 0.706i)5-s + (0.177 − 0.684i)6-s + (−0.515 + 0.297i)7-s − 0.353i·8-s + (−0.0162 − 0.999i)9-s − 0.998·10-s + (0.983 − 0.567i)11-s + (−0.133 − 0.481i)12-s + (0.249 + 0.968i)13-s + (−0.210 + 0.364i)14-s + (−1.36 + 0.376i)15-s + (−0.125 − 0.216i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0806 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.0806 + 0.996i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ -0.0806 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18683 - 1.28670i\)
\(L(\frac12)\) \(\approx\) \(1.18683 - 1.28670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-1.21 + 1.23i)T \)
13 \( 1 + (-0.899 - 3.49i)T \)
good5 \( 1 + (2.73 + 1.57i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.36 - 0.787i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.26 + 1.88i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.06T + 17T^{2} \)
19 \( 1 - 3.76iT - 19T^{2} \)
23 \( 1 + (1.84 - 3.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.109 - 0.189i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.65 - 1.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.292iT - 37T^{2} \)
41 \( 1 + (6.39 + 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.17 + 3.56i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (-9.04 - 5.22i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.00 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.33 + 3.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.772iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.314 - 0.181i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.06iT - 89T^{2} \)
97 \( 1 + (0.535 - 0.309i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.02145123408767951084345167380, −11.62300648205128563289799118974, −9.870630339865191437620442281908, −8.858525895562084723612416192837, −8.016544774501680428610182147196, −6.89670210987759316544815688795, −5.74444409714624092867622211688, −3.99428765228656252818037612548, −3.37269240707416043148735516237, −1.34795659634435179667061789313, 3.09207177273390438861760588219, 3.69978281510907938643427472525, 4.82240093081748244250191274566, 6.44788792239047156685714344617, 7.55149038646452515842131844957, 8.192040129866569188071760344769, 9.603458842236166210586608560164, 10.51270363051937595105065219152, 11.53976661376087895129087495304, 12.44627114308290379150557737650

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