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Properties

Label 2-230-5.3-c2-0-0
Degree $2$
Conductor $230$
Sign $-0.712 - 0.701i$
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 + (−2.41 − 2.41i)3-s − 2i·4-s + (−4.87 − 1.11i)5-s + 4.83·6-s + (3.71 − 3.71i)7-s + (2 + 2i)8-s + 2.69i·9-s + (5.98 − 3.76i)10-s + 5.66·11-s + (−4.83 + 4.83i)12-s + (−11.2 − 11.2i)13-s + 7.43i·14-s + (9.10 + 14.4i)15-s − 4·16-s + (−19.5 + 19.5i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.806 − 0.806i)3-s − 0.5i·4-s + (−0.975 − 0.222i)5-s + 0.806·6-s + (0.531 − 0.531i)7-s + (0.250 + 0.250i)8-s + 0.299i·9-s + (0.598 − 0.376i)10-s + 0.514·11-s + (−0.403 + 0.403i)12-s + (−0.866 − 0.866i)13-s + 0.531i·14-s + (0.606 + 0.964i)15-s − 0.250·16-s + (−1.14 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.712 - 0.701i$
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: \((2,\ 230,\ (\ :1),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0266116 + 0.0649719i\)
\(L(\frac12)\) \(\approx\) \(0.0266116 + 0.0649719i\)
\(L(2)\) 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 \( 1 + (4.87 + 1.11i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 + (2.41 + 2.41i)T + 9iT^{2} \)
7 \( 1 + (-3.71 + 3.71i)T - 49iT^{2} \)
11 \( 1 - 5.66T + 121T^{2} \)
13 \( 1 + (11.2 + 11.2i)T + 169iT^{2} \)
17 \( 1 + (19.5 - 19.5i)T - 289iT^{2} \)
19 \( 1 - 21.5iT - 361T^{2} \)
29 \( 1 - 38.9iT - 841T^{2} \)
31 \( 1 + 39.9T + 961T^{2} \)
37 \( 1 + (-34.6 + 34.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 65.8T + 1.68e3T^{2} \)
43 \( 1 + (35.5 + 35.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (45.5 - 45.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (33.1 + 33.1i)T + 2.80e3iT^{2} \)
59 \( 1 + 32.5iT - 3.48e3T^{2} \)
61 \( 1 + 17.0T + 3.72e3T^{2} \)
67 \( 1 + (15.3 - 15.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 38.4T + 5.04e3T^{2} \)
73 \( 1 + (14.7 + 14.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 32.7iT - 6.24e3T^{2} \)
83 \( 1 + (1.79 + 1.79i)T + 6.88e3iT^{2} \)
89 \( 1 - 130. iT - 7.92e3T^{2} \)
97 \( 1 + (-46.8 + 46.8i)T - 9.40e3iT^{2} \)
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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

−12.45803156894173495826092535309, −11.22339747040833969462429730372, −10.73868057702006457009144640560, −9.235081224449016934839221364155, −8.041201646168817562450593709683, −7.42211716141744881907417832469, −6.47540149888438471248470228883, −5.30288298106980907287523797094, −3.95911148792622325999101123863, −1.39828553379155388028933646208, 0.05225337435721032725146381335, 2.50695050136434992933308260250, 4.29479187689778858161356694202, 4.87540576108694859127951067177, 6.64361108267422064457932193997, 7.70122509078258814626864240135, 8.970645852010227753293421128025, 9.677253510370245605680634598195, 11.01496523804758041116471623740, 11.47931325900689150431321080379

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