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Properties

Label 2-2624-328.157-c0-0-1
Degree $2$
Conductor $2624$
Sign $0.967 - 0.250i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
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  + (0.0600 − 0.144i)3-s + (0.689 + 0.689i)9-s + (1.01 − 0.243i)11-s + (−0.965 + 1.57i)17-s + (1.15 − 0.987i)19-s + (−0.951 + 0.309i)25-s + (0.286 − 0.118i)27-s + (0.0256 − 0.161i)33-s + (0.987 − 0.156i)41-s + (−1.69 + 0.863i)43-s + (0.987 + 0.156i)49-s + (0.170 + 0.234i)51-s + (−0.0737 − 0.226i)57-s + (1.34 + 0.437i)59-s + (0.453 − 1.89i)67-s + ⋯
L(s)  = 1  + (0.0600 − 0.144i)3-s + (0.689 + 0.689i)9-s + (1.01 − 0.243i)11-s + (−0.965 + 1.57i)17-s + (1.15 − 0.987i)19-s + (−0.951 + 0.309i)25-s + (0.286 − 0.118i)27-s + (0.0256 − 0.161i)33-s + (0.987 − 0.156i)41-s + (−1.69 + 0.863i)43-s + (0.987 + 0.156i)49-s + (0.170 + 0.234i)51-s + (−0.0737 − 0.226i)57-s + (1.34 + 0.437i)59-s + (0.453 − 1.89i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $0.967 - 0.250i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (1633, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2624,\ (\ :0),\ 0.967 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.374934954\)
\(L(\frac12)\) \(\approx\) \(1.374934954\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (-0.987 + 0.156i)T \)
good3 \( 1 + (-0.0600 + 0.144i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.987 - 0.156i)T^{2} \)
11 \( 1 + (-1.01 + 0.243i)T + (0.891 - 0.453i)T^{2} \)
13 \( 1 + (0.156 + 0.987i)T^{2} \)
17 \( 1 + (0.965 - 1.57i)T + (-0.453 - 0.891i)T^{2} \)
19 \( 1 + (-1.15 + 0.987i)T + (0.156 - 0.987i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.453 - 0.891i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (1.69 - 0.863i)T + (0.587 - 0.809i)T^{2} \)
47 \( 1 + (-0.987 + 0.156i)T^{2} \)
53 \( 1 + (-0.453 + 0.891i)T^{2} \)
59 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.587 - 0.809i)T^{2} \)
67 \( 1 + (-0.453 + 1.89i)T + (-0.891 - 0.453i)T^{2} \)
71 \( 1 + (0.891 - 0.453i)T^{2} \)
73 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + 1.17iT - T^{2} \)
89 \( 1 + (0.152 - 1.93i)T + (-0.987 - 0.156i)T^{2} \)
97 \( 1 + (-0.303 + 1.26i)T + (-0.891 - 0.453i)T^{2} \)
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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

−9.095239880948562208830552455656, −8.303333601112005974651677059785, −7.57054997198329798234169735146, −6.78506270513352949746603144432, −6.13989195650840048726338298517, −5.13363060946673576277590259570, −4.26925329784815569260897927341, −3.54890298736290294594195829736, −2.25496748131626789499865708180, −1.34216396281428064912484892966, 1.08334614076416637415692934825, 2.29869643635782235375271154065, 3.53880795448025083170926682504, 4.12851374765574401151497576598, 5.06201697087885918775925213710, 5.98235728100548111845277726248, 6.92766687767206544942533654869, 7.25262823943003433107880516846, 8.360875715985542769761655674435, 9.171225163118820646671392688823

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