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Properties

Label 2-230-5.4-c5-0-16
Degree $2$
Conductor $230$
Sign $-0.796 + 0.604i$
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 + 24.7i·3-s − 16·4-s + (44.5 − 33.7i)5-s − 98.8·6-s + 141. i·7-s − 64i·8-s − 367.·9-s + (135. + 178. i)10-s + 750.·11-s − 395. i·12-s + 485. i·13-s − 564.·14-s + (834. + 1.10e3i)15-s + 256·16-s + 1.86e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.58i·3-s − 0.5·4-s + (0.796 − 0.604i)5-s − 1.12·6-s + 1.08i·7-s − 0.353i·8-s − 1.51·9-s + (0.427 + 0.563i)10-s + 1.86·11-s − 0.792i·12-s + 0.796i·13-s − 0.769·14-s + (0.958 + 1.26i)15-s + 0.250·16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.796 + 0.604i$
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: \((2,\ 230,\ (\ :5/2),\ -0.796 + 0.604i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.957543369\)
\(L(\frac12)\) \(\approx\) \(1.957543369\)
\(L(\frac{7}{2})\) 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 \( 1 + (-44.5 + 33.7i)T \)
23 \( 1 + 529iT \)
good3 \( 1 - 24.7iT - 243T^{2} \)
7 \( 1 - 141. iT - 1.68e4T^{2} \)
11 \( 1 - 750.T + 1.61e5T^{2} \)
13 \( 1 - 485. iT - 3.71e5T^{2} \)
17 \( 1 - 1.86e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.59e3T + 2.47e6T^{2} \)
29 \( 1 + 7.41e3T + 2.05e7T^{2} \)
31 \( 1 - 3.60e3T + 2.86e7T^{2} \)
37 \( 1 - 7.42e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.31e4T + 1.15e8T^{2} \)
43 \( 1 + 8.83e3iT - 1.47e8T^{2} \)
47 \( 1 - 708. iT - 2.29e8T^{2} \)
53 \( 1 - 1.02e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.94e4T + 7.14e8T^{2} \)
61 \( 1 + 8.12e3T + 8.44e8T^{2} \)
67 \( 1 + 5.21e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.66e4T + 1.80e9T^{2} \)
73 \( 1 + 6.19e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.53e3T + 3.07e9T^{2} \)
83 \( 1 - 4.47e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.76e4T + 5.58e9T^{2} \)
97 \( 1 + 1.02e5iT - 8.58e9T^{2} \)
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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

−11.89832723798878392911732967348, −10.69790431558574079387230988971, −9.628314849412832680615243881513, −8.954639016374257223651053608951, −8.619655541234552477741143687012, −6.35856373818826400755804572643, −5.85471711005802336773834421728, −4.54558832577067120724909786695, −3.90688951647368499501953562806, −1.85288996638184384176082772067, 0.58267051499636276176991724969, 1.50012641292914111679695834066, 2.59936716976588875849708496218, 4.01678649044382469647042685047, 5.86969929444557827873036272726, 6.82494079396261866889193824585, 7.46230755861875860535216310827, 8.877458051874811304155421164282, 9.864400860398381990460799746400, 11.01986414193402930613494152793

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