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Properties

Label 2-3780-63.58-c1-0-24
Degree $2$
Conductor $3780$
Sign $-0.714 + 0.699i$
Analytic cond. $30.1834$
Root an. cond. $5.49394$
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  + (0.5 + 0.866i)5-s + (−1.66 + 2.05i)7-s + (−2.18 + 3.78i)11-s + (0.772 − 1.33i)13-s + (−3.69 − 6.39i)17-s + (2.20 − 3.81i)19-s + (1.34 + 2.33i)23-s + (−0.499 + 0.866i)25-s + (2.35 + 4.07i)29-s − 0.780·31-s + (−2.61 − 0.417i)35-s + (−1.80 + 3.11i)37-s + (−5.18 + 8.97i)41-s + (2.24 + 3.88i)43-s − 9.71·47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.630 + 0.776i)7-s + (−0.659 + 1.14i)11-s + (0.214 − 0.371i)13-s + (−0.895 − 1.55i)17-s + (0.505 − 0.876i)19-s + (0.280 + 0.486i)23-s + (−0.0999 + 0.173i)25-s + (0.437 + 0.757i)29-s − 0.140·31-s + (−0.441 − 0.0705i)35-s + (−0.296 + 0.512i)37-s + (−0.809 + 1.40i)41-s + (0.341 + 0.591i)43-s − 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3780\)    =    \(2^{2} \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(30.1834\)
Root analytic conductor: \(5.49394\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3780} (3061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3780,\ (\ :1/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1031664376\)
\(L(\frac12)\) \(\approx\) \(0.1031664376\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.66 - 2.05i)T \)
good11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.772 + 1.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.69 + 6.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 3.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.34 - 2.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.35 - 4.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.780T + 31T^{2} \)
37 \( 1 + (1.80 - 3.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.18 - 8.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.71T + 47T^{2} \)
53 \( 1 + (3.60 + 6.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.13T + 59T^{2} \)
61 \( 1 - 0.193T + 61T^{2} \)
67 \( 1 - 9.25T + 67T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (5.37 + 9.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (-1.84 - 3.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.56 + 13.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.22 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.279226674292854821254859636122, −7.25058668777463687512268637223, −6.89228139088930318881544083822, −6.06202944705329363979281541921, −5.00697136070676615868935325473, −4.76983403785919803988205990755, −3.11282922899318186253543699614, −2.84854522499739314909932836319, −1.74431643459146326150210367588, −0.03019150020863908142928449031, 1.20645850611744678454991741109, 2.35516705673813432447479053920, 3.53499275912272461252224582701, 4.00408707974252546058219287597, 5.05725095263775525150509687514, 5.97856945532829989264300066967, 6.38869739969884110380941902263, 7.32354981304557763357363578501, 8.209061949855300621475498782051, 8.642745534370680702382926148125

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