Properties

Label 2-230-23.2-c1-0-7
Degree $2$
Conductor $230$
Sign $-0.991 + 0.131i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (−0.841 − 0.540i)2-s + (−0.420 − 2.92i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (−1.22 + 2.68i)6-s + (2.73 − 3.15i)7-s + (0.142 − 0.989i)8-s + (−5.51 + 1.61i)9-s + (0.654 + 0.755i)10-s + (0.283 − 0.181i)11-s + (2.48 − 1.59i)12-s + (−2.21 − 2.55i)13-s + (−4.00 + 1.17i)14-s + (−0.420 + 2.92i)15-s + (−0.654 + 0.755i)16-s + (−1.82 + 4.00i)17-s + ⋯
L(s)  = 1  + (−0.594 − 0.382i)2-s + (−0.242 − 1.68i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (−0.501 + 1.09i)6-s + (1.03 − 1.19i)7-s + (0.0503 − 0.349i)8-s + (−1.83 + 0.539i)9-s + (0.207 + 0.238i)10-s + (0.0853 − 0.0548i)11-s + (0.718 − 0.461i)12-s + (−0.614 − 0.709i)13-s + (−1.07 + 0.314i)14-s + (−0.108 + 0.755i)15-s + (−0.163 + 0.188i)16-s + (−0.443 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.991 + 0.131i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0496074 - 0.753970i\)
\(L(\frac12)\) \(\approx\) \(0.0496074 - 0.753970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (2.79 - 3.89i)T \)
good3 \( 1 + (0.420 + 2.92i)T + (-2.87 + 0.845i)T^{2} \)
7 \( 1 + (-2.73 + 3.15i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-0.283 + 0.181i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.21 + 2.55i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (1.82 - 4.00i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-2.00 - 4.39i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-3.04 + 6.67i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.759 + 5.28i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-7.80 + 2.29i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-0.0346 - 0.0101i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (1.80 + 12.5i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 2.78T + 47T^{2} \)
53 \( 1 + (0.0676 - 0.0780i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (5.00 + 5.77i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.30 - 9.06i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-7.18 - 4.62i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-12.6 - 8.12i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (2.21 + 4.85i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-8.98 - 10.3i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (0.0754 - 0.0221i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.30 + 9.08i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (3.45 + 1.01i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.76179846710063799431243742020, −11.03537823210879181735011804110, −9.990524501299087952987020222857, −8.181404783531351038104572510130, −7.889876570728448873951827528986, −7.11660990802739391955752821860, −5.79750748909259443031868996903, −4.01803361250208661931879739361, −2.05789050294137810859155999163, −0.801115124181985169879200970705, 2.73824740973624178002003074009, 4.68689447235018757151225930208, 5.03254553173133636974224381860, 6.54617245650447114792704892939, 8.046727879007272242815447194990, 9.043736939668890143920653592079, 9.494954438142335907282442200273, 10.76226219376225094219723066526, 11.40352474728181780676613947304, 12.10858583123265403091910780545

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