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Properties

Label 2-230-5.2-c2-0-1
Degree $2$
Conductor $230$
Sign $-0.339 + 0.940i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (−2.98 + 2.98i)3-s + 2i·4-s + (−3.91 + 3.11i)5-s − 5.97·6-s + (−3.94 − 3.94i)7-s + (−2 + 2i)8-s − 8.83i·9-s + (−7.02 − 0.804i)10-s + 8.59·11-s + (−5.97 − 5.97i)12-s + (2.89 − 2.89i)13-s − 7.89i·14-s + (2.40 − 20.9i)15-s − 4·16-s + (−15.7 − 15.7i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.995 + 0.995i)3-s + 0.5i·4-s + (−0.782 + 0.622i)5-s − 0.995·6-s + (−0.563 − 0.563i)7-s + (−0.250 + 0.250i)8-s − 0.981i·9-s + (−0.702 − 0.0804i)10-s + 0.781·11-s + (−0.497 − 0.497i)12-s + (0.223 − 0.223i)13-s − 0.563i·14-s + (0.160 − 1.39i)15-s − 0.250·16-s + (−0.924 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.339 + 0.940i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.339 + 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.177882 - 0.253177i\)
\(L(\frac12)\) \(\approx\) \(0.177882 - 0.253177i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (3.91 - 3.11i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good3 \( 1 + (2.98 - 2.98i)T - 9iT^{2} \)
7 \( 1 + (3.94 + 3.94i)T + 49iT^{2} \)
11 \( 1 - 8.59T + 121T^{2} \)
13 \( 1 + (-2.89 + 2.89i)T - 169iT^{2} \)
17 \( 1 + (15.7 + 15.7i)T + 289iT^{2} \)
19 \( 1 - 12.0iT - 361T^{2} \)
29 \( 1 - 41.1iT - 841T^{2} \)
31 \( 1 + 19.8T + 961T^{2} \)
37 \( 1 + (40.5 + 40.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.8T + 1.68e3T^{2} \)
43 \( 1 + (29.7 - 29.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-5.79 - 5.79i)T + 2.20e3iT^{2} \)
53 \( 1 + (8.37 - 8.37i)T - 2.80e3iT^{2} \)
59 \( 1 - 91.8iT - 3.48e3T^{2} \)
61 \( 1 + 6.02T + 3.72e3T^{2} \)
67 \( 1 + (-21.2 - 21.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 89.1T + 5.04e3T^{2} \)
73 \( 1 + (80.4 - 80.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 66.7iT - 6.24e3T^{2} \)
83 \( 1 + (-30.2 + 30.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 48.7iT - 7.92e3T^{2} \)
97 \( 1 + (40.6 + 40.6i)T + 9.40e3iT^{2} \)
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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

−12.45306139050342747621288013380, −11.56620544383270800447016878594, −10.88069458025905995303500935844, −9.996214112997752460983111106120, −8.763309340703069515146712002691, −7.23901962573783045172710473120, −6.56286740361817798195641033866, −5.34047942343498362043076673195, −4.19884465166487516666819498845, −3.43565623860273548095872115305, 0.15900814042885706803916943317, 1.74240386030838750138306004550, 3.71229592350848740721771837765, 4.98525952425706930354785592573, 6.20439486472081850712326356162, 6.84223594949996296658889835778, 8.341450565212522687744642237938, 9.357113630161835297299428331506, 10.84589516971355161894724099239, 11.73010837430878717656852002648

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