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Properties

Label 2-3900-65.18-c1-0-18
Degree $2$
Conductor $3900$
Sign $0.315 - 0.949i$
Analytic cond. $31.1416$
Root an. cond. $5.58047$
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  + (0.707 + 0.707i)3-s − 1.09·7-s + 1.00i·9-s + (2.73 + 2.73i)11-s + (2.92 − 2.10i)13-s + (4.68 + 4.68i)17-s + (−3.07 − 3.07i)19-s + (−0.773 − 0.773i)21-s + (−0.442 + 0.442i)23-s + (−0.707 + 0.707i)27-s + 8.91i·29-s + (3.54 − 3.54i)31-s + 3.87i·33-s − 5.84·37-s + (3.55 + 0.580i)39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s − 0.413·7-s + 0.333i·9-s + (0.825 + 0.825i)11-s + (0.811 − 0.584i)13-s + (1.13 + 1.13i)17-s + (−0.705 − 0.705i)19-s + (−0.168 − 0.168i)21-s + (−0.0923 + 0.0923i)23-s + (−0.136 + 0.136i)27-s + 1.65i·29-s + (0.636 − 0.636i)31-s + 0.674i·33-s − 0.960·37-s + (0.569 + 0.0929i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.315 - 0.949i$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (2293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ 0.315 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.240346812\)
\(L(\frac12)\) \(\approx\) \(2.240346812\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
13 \( 1 + (-2.92 + 2.10i)T \)
good7 \( 1 + 1.09T + 7T^{2} \)
11 \( 1 + (-2.73 - 2.73i)T + 11iT^{2} \)
17 \( 1 + (-4.68 - 4.68i)T + 17iT^{2} \)
19 \( 1 + (3.07 + 3.07i)T + 19iT^{2} \)
23 \( 1 + (0.442 - 0.442i)T - 23iT^{2} \)
29 \( 1 - 8.91iT - 29T^{2} \)
31 \( 1 + (-3.54 + 3.54i)T - 31iT^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + (-6.59 + 6.59i)T - 41iT^{2} \)
43 \( 1 + (5.19 - 5.19i)T - 43iT^{2} \)
47 \( 1 + 1.17T + 47T^{2} \)
53 \( 1 + (-1.87 - 1.87i)T + 53iT^{2} \)
59 \( 1 + (-8.13 + 8.13i)T - 59iT^{2} \)
61 \( 1 - 9.80T + 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + (-0.984 + 0.984i)T - 71iT^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 - 1.52T + 83T^{2} \)
89 \( 1 + (4.35 - 4.35i)T - 89iT^{2} \)
97 \( 1 - 7.73iT - 97T^{2} \)
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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

−8.539735083472511032363024430537, −8.122959267959744800500081686089, −7.07780805819376745139155667287, −6.48422624249597061941233351545, −5.62973651048903842873171708683, −4.79322481454534067475669201496, −3.79257619631150021047052301305, −3.42271237734652959695596353856, −2.21133393766809423480415235582, −1.16244673037359735497004432252, 0.69705794263544283845271881654, 1.70622289133384668463253084318, 2.87684211439364258253601428589, 3.59738161723458273371929884319, 4.31850588370146024944025626553, 5.54658732867361179104231583161, 6.23744047809148872764800483921, 6.76891978421435095193671138325, 7.64388433150667976752255004421, 8.447709589139226804338955542295

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