Properties

Label 2-624-39.11-c1-0-11
Degree $2$
Conductor $624$
Sign $0.399 - 0.916i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
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  + (1.60 + 0.650i)3-s + (−1.06 + 1.06i)5-s + (−0.366 − 1.36i)7-s + (2.15 + 2.08i)9-s + (−1.06 + 3.97i)11-s + (3.59 + 0.232i)13-s + (−2.40 + 1.01i)15-s + (−2.51 + 4.36i)17-s + (3.73 − i)19-s + (0.301 − 2.43i)21-s + 2.73i·25-s + (2.09 + 4.75i)27-s + (6.20 − 3.58i)29-s + (2.46 + 2.46i)31-s + (−4.29 + 5.68i)33-s + ⋯
L(s)  = 1  + (0.926 + 0.375i)3-s + (−0.476 + 0.476i)5-s + (−0.138 − 0.516i)7-s + (0.717 + 0.696i)9-s + (−0.321 + 1.19i)11-s + (0.997 + 0.0643i)13-s + (−0.620 + 0.262i)15-s + (−0.611 + 1.05i)17-s + (0.856 − 0.229i)19-s + (0.0657 − 0.530i)21-s + 0.546i·25-s + (0.403 + 0.914i)27-s + (1.15 − 0.665i)29-s + (0.442 + 0.442i)31-s + (−0.747 + 0.989i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54456 + 1.01190i\)
\(L(\frac12)\) \(\approx\) \(1.54456 + 1.01190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 0.650i)T \)
13 \( 1 + (-3.59 - 0.232i)T \)
good5 \( 1 + (1.06 - 1.06i)T - 5iT^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.51 - 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.73 + i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.20 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.46 - 2.46i)T + 31iT^{2} \)
37 \( 1 + (5.23 + 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.42 + 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.25 + 4.25i)T + 47iT^{2} \)
53 \( 1 - 0.779iT - 53T^{2} \)
59 \( 1 + (2.90 - 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.901 - 0.901i)T - 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (-2.90 + 2.90i)T - 83iT^{2} \)
89 \( 1 + (-2.41 + 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.63 + 0.437i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.47939420524326481261163801615, −10.07478356234831459784101518630, −8.959364517478074644729925542910, −8.171235893161427204018040738999, −7.31823528824903481013572716937, −6.58302418498085128061765540739, −4.97422360725413376966853519876, −3.99638463609168009833063419066, −3.22018295799705620681669673962, −1.81707856200182267366479567965, 1.00284354728127132521567759653, 2.71955314992746300220660379646, 3.53118871726431835882892173921, 4.79988950026954606932974704777, 6.01678013140212082717783543351, 6.97929062019292537342013726897, 8.160356328924327065112402642172, 8.534208019817588966557479915394, 9.287291684778882537588992389289, 10.36514189505061847516592499264

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