Properties

Label 2-600-40.29-c1-0-35
Degree $2$
Conductor $600$
Sign $-0.316 + 0.948i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + 3-s − 2i·4-s + (1 − i)6-s − 2i·7-s + (−2 − 2i)8-s + 9-s − 4i·11-s − 2i·12-s + (−2 − 2i)14-s − 4·16-s + 6i·17-s + (1 − i)18-s + 4i·19-s − 2i·21-s + (−4 − 4i)22-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + 0.577·3-s i·4-s + (0.408 − 0.408i)6-s − 0.755i·7-s + (−0.707 − 0.707i)8-s + 0.333·9-s − 1.20i·11-s − 0.577i·12-s + (−0.534 − 0.534i)14-s − 16-s + 1.45i·17-s + (0.235 − 0.235i)18-s + 0.917i·19-s − 0.436i·21-s + (−0.852 − 0.852i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43508 - 1.99107i\)
\(L(\frac12)\) \(\approx\) \(1.43508 - 1.99107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.46558207188909209278887373858, −9.862013650422922002899832201568, −8.640995296333488094638450261966, −7.937331385299339575120187278410, −6.50607490609173362501205439226, −5.83440828607214858299831450367, −4.36377315028797765063199726698, −3.72465792120125972765801956572, −2.60573082288179315085877908054, −1.12033754783572824717961723555, 2.30106681279212984081645907654, 3.22959532377526981731033881577, 4.63689501371715664178488111216, 5.20591690215189884082556029488, 6.58055249715388620260134079843, 7.25484778424901926075178841069, 8.129832662337429748606819928419, 9.138990688243248336757955051629, 9.662488699462513012352011175620, 11.17280213434441775729354164527

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