Properties

Label 2-650-5.4-c5-0-27
Degree $2$
Conductor $650$
Sign $-0.447 - 0.894i$
Analytic cond. $104.249$
Root an. cond. $10.2102$
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 − 3.71i·3-s − 16·4-s + 14.8·6-s + 10.2i·7-s − 64i·8-s + 229.·9-s + 197.·11-s + 59.4i·12-s + 169i·13-s − 41.0·14-s + 256·16-s + 949. i·17-s + 916. i·18-s − 2.23e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.238i·3-s − 0.5·4-s + 0.168·6-s + 0.0791i·7-s − 0.353i·8-s + 0.943·9-s + 0.491·11-s + 0.119i·12-s + 0.277i·13-s − 0.0559·14-s + 0.250·16-s + 0.797i·17-s + 0.666i·18-s − 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(104.249\)
Root analytic conductor: \(10.2102\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :5/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.803941768\)
\(L(\frac12)\) \(\approx\) \(1.803941768\)
\(L(\frac{7}{2})\) 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 \( 1 \)
13 \( 1 - 169iT \)
good3 \( 1 + 3.71iT - 243T^{2} \)
7 \( 1 - 10.2iT - 1.68e4T^{2} \)
11 \( 1 - 197.T + 1.61e5T^{2} \)
17 \( 1 - 949. iT - 1.41e6T^{2} \)
19 \( 1 + 2.23e3T + 2.47e6T^{2} \)
23 \( 1 + 367. iT - 6.43e6T^{2} \)
29 \( 1 - 6.60e3T + 2.05e7T^{2} \)
31 \( 1 + 9.91e3T + 2.86e7T^{2} \)
37 \( 1 - 9.39e3iT - 6.93e7T^{2} \)
41 \( 1 - 2.05e4T + 1.15e8T^{2} \)
43 \( 1 + 1.63e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.35e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.50e4iT - 4.18e8T^{2} \)
59 \( 1 - 6.17e3T + 7.14e8T^{2} \)
61 \( 1 - 1.37e4T + 8.44e8T^{2} \)
67 \( 1 - 6.21e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.83e4T + 1.80e9T^{2} \)
73 \( 1 - 5.99e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.96e4T + 3.07e9T^{2} \)
83 \( 1 + 3.81e4iT - 3.93e9T^{2} \)
89 \( 1 - 7.82e4T + 5.58e9T^{2} \)
97 \( 1 + 8.05e4iT - 8.58e9T^{2} \)
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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

−10.01490965016339175430817571516, −8.963840970674933210702280936329, −8.327843018468920554763025387653, −7.25022698745281790057090244978, −6.61958254984071574936630875872, −5.77717998378022361716435020857, −4.49463412409470356696621570470, −3.86125847059985752723520927594, −2.17883960659460888311907623577, −1.02559934351973872989535480549, 0.44409259208751514494333250563, 1.58768049422526518156796549212, 2.71539693346077122485768156496, 3.96658780277569106644300777851, 4.54960975823399968666830598478, 5.76425401034089152965326611079, 6.90333102816494350149405095376, 7.79426425534590172251355276095, 8.994315736877392583497952689137, 9.484984370860072076282322543736

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