Properties

Label 2-230-115.19-c2-0-23
Degree $2$
Conductor $230$
Sign $-0.994 + 0.104i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 − 0.587i)2-s + (−1.62 − 5.54i)3-s + (1.30 − 1.51i)4-s + (0.750 − 4.94i)5-s + (−5.35 − 6.17i)6-s + (−0.543 − 3.77i)7-s + (0.796 − 2.71i)8-s + (−20.5 + 13.1i)9-s + (−1.93 − 6.80i)10-s + (15.1 + 6.92i)11-s + (−10.5 − 4.80i)12-s + (6.09 + 0.876i)13-s + (−2.91 − 4.54i)14-s + (−28.6 + 3.88i)15-s + (−0.569 − 3.95i)16-s + (8.15 + 9.40i)17-s + ⋯
L(s)  = 1  + (0.643 − 0.293i)2-s + (−0.542 − 1.84i)3-s + (0.327 − 0.377i)4-s + (0.150 − 0.988i)5-s + (−0.892 − 1.02i)6-s + (−0.0775 − 0.539i)7-s + (0.0996 − 0.339i)8-s + (−2.28 + 1.46i)9-s + (−0.193 − 0.680i)10-s + (1.37 + 0.629i)11-s + (−0.876 − 0.400i)12-s + (0.468 + 0.0674i)13-s + (−0.208 − 0.324i)14-s + (−1.90 + 0.259i)15-s + (−0.0355 − 0.247i)16-s + (0.479 + 0.553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.994 + 0.104i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.994 + 0.104i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0994140 - 1.90011i\)
\(L(\frac12)\) \(\approx\) \(0.0994140 - 1.90011i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.587i)T \)
5 \( 1 + (-0.750 + 4.94i)T \)
23 \( 1 + (-1.00 + 22.9i)T \)
good3 \( 1 + (1.62 + 5.54i)T + (-7.57 + 4.86i)T^{2} \)
7 \( 1 + (0.543 + 3.77i)T + (-47.0 + 13.8i)T^{2} \)
11 \( 1 + (-15.1 - 6.92i)T + (79.2 + 91.4i)T^{2} \)
13 \( 1 + (-6.09 - 0.876i)T + (162. + 47.6i)T^{2} \)
17 \( 1 + (-8.15 - 9.40i)T + (-41.1 + 286. i)T^{2} \)
19 \( 1 + (-18.5 - 16.1i)T + (51.3 + 357. i)T^{2} \)
29 \( 1 + (26.1 + 30.1i)T + (-119. + 832. i)T^{2} \)
31 \( 1 + (37.5 + 11.0i)T + (808. + 519. i)T^{2} \)
37 \( 1 + (17.7 - 11.4i)T + (568. - 1.24e3i)T^{2} \)
41 \( 1 + (-29.7 - 19.0i)T + (698. + 1.52e3i)T^{2} \)
43 \( 1 + (3.53 - 1.03i)T + (1.55e3 - 9.99e2i)T^{2} \)
47 \( 1 - 20.2iT - 2.20e3T^{2} \)
53 \( 1 + (2.70 + 18.7i)T + (-2.69e3 + 791. i)T^{2} \)
59 \( 1 + (7.82 - 54.4i)T + (-3.33e3 - 980. i)T^{2} \)
61 \( 1 + (-27.3 + 93.2i)T + (-3.13e3 - 2.01e3i)T^{2} \)
67 \( 1 + (-30.8 - 67.4i)T + (-2.93e3 + 3.39e3i)T^{2} \)
71 \( 1 + (-15.0 - 33.0i)T + (-3.30e3 + 3.80e3i)T^{2} \)
73 \( 1 + (22.4 + 19.4i)T + (758. + 5.27e3i)T^{2} \)
79 \( 1 + (-83.0 - 11.9i)T + (5.98e3 + 1.75e3i)T^{2} \)
83 \( 1 + (12.2 - 7.84i)T + (2.86e3 - 6.26e3i)T^{2} \)
89 \( 1 + (-31.9 - 108. i)T + (-6.66e3 + 4.28e3i)T^{2} \)
97 \( 1 + (142. + 91.6i)T + (3.90e3 + 8.55e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.96694557081278941940173367515, −11.07174600165806388320136088325, −9.579782292886093034751531083160, −8.249775211623786748727415657487, −7.28142570777955758522560945442, −6.28775045237999868919457721294, −5.49665906838221728192338440838, −3.96786026389108908449032423999, −1.85453924298846892218452780004, −0.986563200830738821651045690561, 3.22551789406868649001961192708, 3.76217967682705799662649361519, 5.30511663539481518759914218422, 5.88212405733844908033039587229, 7.08246674107666282316441217672, 8.996769252442583912850069693749, 9.475950735815975675130335116863, 10.80400404822701549567775297434, 11.32629707096471399832662013320, 12.02706643009763440691105383623

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