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Properties

Label 2-230-5.3-c2-0-1
Degree $2$
Conductor $230$
Sign $0.379 - 0.925i$
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 + (−3.78 − 3.78i)3-s − 2i·4-s + (0.818 + 4.93i)5-s − 7.57·6-s + (−3.70 + 3.70i)7-s + (−2 − 2i)8-s + 19.7i·9-s + (5.75 + 4.11i)10-s − 13.1·11-s + (−7.57 + 7.57i)12-s + (6.65 + 6.65i)13-s + 7.40i·14-s + (15.5 − 21.7i)15-s − 4·16-s + (8.90 − 8.90i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−1.26 − 1.26i)3-s − 0.5i·4-s + (0.163 + 0.986i)5-s − 1.26·6-s + (−0.529 + 0.529i)7-s + (−0.250 − 0.250i)8-s + 2.19i·9-s + (0.575 + 0.411i)10-s − 1.19·11-s + (−0.631 + 0.631i)12-s + (0.511 + 0.511i)13-s + 0.529i·14-s + (1.03 − 1.45i)15-s − 0.250·16-s + (0.523 − 0.523i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.402858 + 0.270196i\)
\(L(\frac12)\) \(\approx\) \(0.402858 + 0.270196i\)
\(L(2)\) 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 \( 1 + (-0.818 - 4.93i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 + (3.78 + 3.78i)T + 9iT^{2} \)
7 \( 1 + (3.70 - 3.70i)T - 49iT^{2} \)
11 \( 1 + 13.1T + 121T^{2} \)
13 \( 1 + (-6.65 - 6.65i)T + 169iT^{2} \)
17 \( 1 + (-8.90 + 8.90i)T - 289iT^{2} \)
19 \( 1 - 10.0iT - 361T^{2} \)
29 \( 1 - 40.4iT - 841T^{2} \)
31 \( 1 + 59.9T + 961T^{2} \)
37 \( 1 + (-11.4 + 11.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 2.32T + 1.68e3T^{2} \)
43 \( 1 + (-12.0 - 12.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (38.4 - 38.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (49.3 + 49.3i)T + 2.80e3iT^{2} \)
59 \( 1 + 23.6iT - 3.48e3T^{2} \)
61 \( 1 + 97.8T + 3.72e3T^{2} \)
67 \( 1 + (-63.3 + 63.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 42.9T + 5.04e3T^{2} \)
73 \( 1 + (9.09 + 9.09i)T + 5.32e3iT^{2} \)
79 \( 1 - 131. iT - 6.24e3T^{2} \)
83 \( 1 + (-76.9 - 76.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 16.9iT - 7.92e3T^{2} \)
97 \( 1 + (41.5 - 41.5i)T - 9.40e3iT^{2} \)
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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

−12.33899698231166616939483820056, −11.09123258798366226915991069258, −10.90285105080238551122849577921, −9.582380129351895642656914985795, −7.76044514517300559149682097437, −6.84803904399523099838109562936, −5.96157001297920515170337214112, −5.24137194658842434447558657178, −3.12091884112315470099703658258, −1.79927175342209506912288594101, 0.25085892984903125466639324426, 3.57758283390684016920437744194, 4.62251923129150041812746625084, 5.46800369111305140141645914414, 6.14968315871534273756536910551, 7.71180122527062367255321518314, 9.022044461308537862910269876122, 10.05016853876434242829151041260, 10.75561840216485614077670225764, 11.81652948043946629358208636771

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