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Properties

Label 2-3825-17.14-c0-0-1
Degree $2$
Conductor $3825$
Sign $0.434 + 0.900i$
Analytic cond. $1.90892$
Root an. cond. $1.38163$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.360 + 0.149i)2-s + (−0.599 + 0.599i)4-s + (0.275 − 0.666i)8-s − 0.566i·16-s + (0.195 − 0.980i)17-s + (−0.707 + 0.292i)19-s + (−0.275 − 1.38i)23-s + (−1.08 − 0.216i)31-s + (0.360 + 0.870i)32-s + (0.0761 + 0.382i)34-s + (0.211 − 0.211i)38-s + (0.306 + 0.458i)46-s + (−1.17 − 1.17i)47-s + (−0.382 − 0.923i)49-s + (1.81 − 0.750i)53-s + ⋯
L(s)  = 1  + (−0.360 + 0.149i)2-s + (−0.599 + 0.599i)4-s + (0.275 − 0.666i)8-s − 0.566i·16-s + (0.195 − 0.980i)17-s + (−0.707 + 0.292i)19-s + (−0.275 − 1.38i)23-s + (−1.08 − 0.216i)31-s + (0.360 + 0.870i)32-s + (0.0761 + 0.382i)34-s + (0.211 − 0.211i)38-s + (0.306 + 0.458i)46-s + (−1.17 − 1.17i)47-s + (−0.382 − 0.923i)49-s + (1.81 − 0.750i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (3601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ 0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6175893966\)
\(L(\frac12)\) \(\approx\) \(0.6175893966\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 + (-0.195 + 0.980i)T \)
good2 \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.275 + 1.38i)T + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
53 \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (0.750 + 1.81i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.382 + 0.923i)T^{2} \)
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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

−8.648968970251408233088006777986, −7.88153962101476533366560740434, −7.17304960844125977116845134995, −6.53573755623174881818465870647, −5.48260577484509157118947492437, −4.68624688664794291511315060536, −3.95452087428937047803451480871, −3.11343663681364616753024096487, −2.03863750553205744893307562249, −0.42957157476239322507773412283, 1.27885230584865318480112484556, 2.12351019316712984751078572689, 3.45249030985894500276647186374, 4.23929593808820404227973089875, 5.09706983963484135987816234754, 5.80731607621068446863044721342, 6.46700698184290630073515726545, 7.54730790124402759219477684972, 8.151023781696305630355109441436, 8.915024066700706033120218712860

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