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Properties

Label 2-230-5.4-c5-0-11
Degree $2$
Conductor $230$
Sign $-0.138 + 0.990i$
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 + 13.9i·3-s − 16·4-s + (−7.76 + 55.3i)5-s − 55.9·6-s + 114. i·7-s − 64i·8-s + 47.4·9-s + (−221. − 31.0i)10-s − 208.·11-s − 223. i·12-s + 757. i·13-s − 458.·14-s + (−774. − 108. i)15-s + 256·16-s + 1.02e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.897i·3-s − 0.5·4-s + (−0.138 + 0.990i)5-s − 0.634·6-s + 0.885i·7-s − 0.353i·8-s + 0.195·9-s + (−0.700 − 0.0982i)10-s − 0.520·11-s − 0.448i·12-s + 1.24i·13-s − 0.625·14-s + (−0.888 − 0.124i)15-s + 0.250·16-s + 0.863i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.138 + 0.990i$
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: \((2,\ 230,\ (\ :5/2),\ -0.138 + 0.990i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.313730459\)
\(L(\frac12)\) \(\approx\) \(1.313730459\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.76 - 55.3i)T \)
23 \( 1 - 529iT \)
good3 \( 1 - 13.9iT - 243T^{2} \)
7 \( 1 - 114. iT - 1.68e4T^{2} \)
11 \( 1 + 208.T + 1.61e5T^{2} \)
13 \( 1 - 757. iT - 3.71e5T^{2} \)
17 \( 1 - 1.02e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.92e3T + 2.47e6T^{2} \)
29 \( 1 - 4.59e3T + 2.05e7T^{2} \)
31 \( 1 - 6.06e3T + 2.86e7T^{2} \)
37 \( 1 + 1.08e4iT - 6.93e7T^{2} \)
41 \( 1 + 3.49e3T + 1.15e8T^{2} \)
43 \( 1 - 6.23e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.09e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.02e4iT - 4.18e8T^{2} \)
59 \( 1 + 5.29e3T + 7.14e8T^{2} \)
61 \( 1 - 5.44e4T + 8.44e8T^{2} \)
67 \( 1 + 3.99e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.37e3T + 1.80e9T^{2} \)
73 \( 1 - 2.57e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.44e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5iT - 3.93e9T^{2} \)
89 \( 1 - 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.04e4iT - 8.58e9T^{2} \)
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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

−11.95913428666074169002153225504, −10.82507986578522730391899895391, −10.08826438885745844779133500340, −9.097315305086695286861350040928, −8.152232280835082731462878296931, −6.85323537895861687412047945912, −6.05583163259470834595667511703, −4.71959300006898749887994110282, −3.76536334517150318786465460719, −2.24761894023787364815523416388, 0.43861555690263882406524283869, 1.12155736866331195464604544912, 2.58942870914127995655778941203, 4.16793943707071797380952613532, 5.15117526880207403971182703930, 6.66446171896278744493164072132, 7.86578465538952560191218584358, 8.452389311260469485897123951155, 9.918338790474418445750953564080, 10.56661087210462673531607883932

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