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Properties

Label 2-230-5.2-c2-0-13
Degree $2$
Conductor $230$
Sign $0.590 + 0.806i$
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 + (2.93 − 2.93i)3-s + 2i·4-s + (0.390 + 4.98i)5-s − 5.86·6-s + (1.49 + 1.49i)7-s + (2 − 2i)8-s − 8.20i·9-s + (4.59 − 5.37i)10-s + 8.24·11-s + (5.86 + 5.86i)12-s + (14.8 − 14.8i)13-s − 2.99i·14-s + (15.7 + 13.4i)15-s − 4·16-s + (8.86 + 8.86i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.977 − 0.977i)3-s + 0.5i·4-s + (0.0781 + 0.996i)5-s − 0.977·6-s + (0.214 + 0.214i)7-s + (0.250 − 0.250i)8-s − 0.912i·9-s + (0.459 − 0.537i)10-s + 0.749·11-s + (0.488 + 0.488i)12-s + (1.14 − 1.14i)13-s − 0.214i·14-s + (1.05 + 0.898i)15-s − 0.250·16-s + (0.521 + 0.521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.590 + 0.806i$
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: \((2,\ 230,\ (\ :1),\ 0.590 + 0.806i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.63100 - 0.827507i\)
\(L(\frac12)\) \(\approx\) \(1.63100 - 0.827507i\)
\(L(2)\) 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 \( 1 + (-0.390 - 4.98i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good3 \( 1 + (-2.93 + 2.93i)T - 9iT^{2} \)
7 \( 1 + (-1.49 - 1.49i)T + 49iT^{2} \)
11 \( 1 - 8.24T + 121T^{2} \)
13 \( 1 + (-14.8 + 14.8i)T - 169iT^{2} \)
17 \( 1 + (-8.86 - 8.86i)T + 289iT^{2} \)
19 \( 1 + 8.33iT - 361T^{2} \)
29 \( 1 + 21.2iT - 841T^{2} \)
31 \( 1 + 38.0T + 961T^{2} \)
37 \( 1 + (-18.9 - 18.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 42.1T + 1.68e3T^{2} \)
43 \( 1 + (52.2 - 52.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (15.0 + 15.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-27.3 + 27.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 40.1iT - 3.48e3T^{2} \)
61 \( 1 - 9.06T + 3.72e3T^{2} \)
67 \( 1 + (8.60 + 8.60i)T + 4.48e3iT^{2} \)
71 \( 1 + 124.T + 5.04e3T^{2} \)
73 \( 1 + (98.5 - 98.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 13.6iT - 6.24e3T^{2} \)
83 \( 1 + (0.140 - 0.140i)T - 6.88e3iT^{2} \)
89 \( 1 - 134. iT - 7.92e3T^{2} \)
97 \( 1 + (82.0 + 82.0i)T + 9.40e3iT^{2} \)
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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

−11.73382954617755698051082121558, −10.90751979580000043090932919362, −9.882864239870531011473876335034, −8.693315134012122026858179670839, −8.005614120694850466249851069628, −7.09008791077227217936027609469, −6.00005846461318226329001883688, −3.65260298107391142959075915561, −2.71582998820405241974473425723, −1.40795323267149424784341240060, 1.50309248183824161410920586117, 3.68574781623609900215216509813, 4.58140741191509803714277994247, 5.92658718298754033277136088594, 7.39103770453809446066686630072, 8.612895325233910616706468045946, 9.039578217735313001992455308365, 9.721133910289711595636037359009, 10.91402123762811553809560290518, 12.01643666156057502164079174651

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