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Properties

Label 2-230-5.4-c5-0-7
Degree $2$
Conductor $230$
Sign $0.207 - 0.978i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 19.5i·3-s − 16·4-s + (11.6 − 54.6i)5-s + 78.2·6-s + 53.5i·7-s − 64i·8-s − 140.·9-s + (218. + 46.5i)10-s − 780.·11-s + 313. i·12-s + 672. i·13-s − 214.·14-s + (−1.07e3 − 227. i)15-s + 256·16-s + 1.46e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.25i·3-s − 0.5·4-s + (0.207 − 0.978i)5-s + 0.887·6-s + 0.413i·7-s − 0.353i·8-s − 0.576·9-s + (0.691 + 0.147i)10-s − 1.94·11-s + 0.627i·12-s + 1.10i·13-s − 0.292·14-s + (−1.22 − 0.261i)15-s + 0.250·16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ 0.207 - 0.978i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.008880356\)
\(L(\frac12)\) \(\approx\) \(1.008880356\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (-11.6 + 54.6i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 19.5iT - 243T^{2} \)
7 \( 1 - 53.5iT - 1.68e4T^{2} \)
11 \( 1 + 780.T + 1.61e5T^{2} \)
13 \( 1 - 672. iT - 3.71e5T^{2} \)
17 \( 1 - 1.46e3iT - 1.41e6T^{2} \)
19 \( 1 - 644.T + 2.47e6T^{2} \)
29 \( 1 - 1.67e3T + 2.05e7T^{2} \)
31 \( 1 - 8.39e3T + 2.86e7T^{2} \)
37 \( 1 + 1.72e3iT - 6.93e7T^{2} \)
41 \( 1 + 3.75e3T + 1.15e8T^{2} \)
43 \( 1 + 5.02e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.24e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.59e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.55e4T + 7.14e8T^{2} \)
61 \( 1 + 4.46e4T + 8.44e8T^{2} \)
67 \( 1 + 6.55e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.81e4T + 1.80e9T^{2} \)
73 \( 1 - 4.03e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.86e4T + 3.07e9T^{2} \)
83 \( 1 - 1.02e5iT - 3.93e9T^{2} \)
89 \( 1 - 1.26e5T + 5.58e9T^{2} \)
97 \( 1 - 3.81e4iT - 8.58e9T^{2} \)
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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

−12.06443580479644509096425662088, −10.49151602634069936946713110968, −9.270694595963396444744326966392, −8.201580888235429021109028496706, −7.73905335258245833030042100051, −6.46200985774855702398738311104, −5.59610444384675947827332568006, −4.50461571814705240102485953252, −2.38986254760675500303443347142, −1.14871049193744847911991809239, 0.32176281236983739750069151532, 2.65134055319139464967337575403, 3.27579309803454183166923316941, 4.72085482578853866738710181996, 5.51118577714260766431820744692, 7.25149868894910192102270336969, 8.306694721461391664839601460642, 9.839877931430513133870645811295, 10.21173623991548562665175203830, 10.75126844006145737971144165096

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