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Properties

Label 2-124-4.3-c2-0-9
Degree $2$
Conductor $124$
Sign $0.971 + 0.236i$
Analytic cond. $3.37875$
Root an. cond. $1.83813$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 1.57i)2-s + 0.706i·3-s + (−0.946 + 3.88i)4-s − 0.787·5-s + (1.11 − 0.873i)6-s − 0.940i·7-s + (7.28 − 3.31i)8-s + 8.50·9-s + (0.972 + 1.23i)10-s + 14.5i·11-s + (−2.74 − 0.669i)12-s + 23.3·13-s + (−1.47 + 1.16i)14-s − 0.556i·15-s + (−14.2 − 7.36i)16-s + 11.2·17-s + ⋯
L(s)  = 1  + (−0.617 − 0.786i)2-s + 0.235i·3-s + (−0.236 + 0.971i)4-s − 0.157·5-s + (0.185 − 0.145i)6-s − 0.134i·7-s + (0.910 − 0.414i)8-s + 0.944·9-s + (0.0972 + 0.123i)10-s + 1.32i·11-s + (−0.228 − 0.0557i)12-s + 1.79·13-s + (−0.105 + 0.0830i)14-s − 0.0371i·15-s + (−0.887 − 0.460i)16-s + 0.662·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ 0.971 + 0.236i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08111 - 0.129815i\)
\(L(\frac12)\) \(\approx\) \(1.08111 - 0.129815i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 1.57i)T \)
31 \( 1 - 5.56iT \)
good3 \( 1 - 0.706iT - 9T^{2} \)
5 \( 1 + 0.787T + 25T^{2} \)
7 \( 1 + 0.940iT - 49T^{2} \)
11 \( 1 - 14.5iT - 121T^{2} \)
13 \( 1 - 23.3T + 169T^{2} \)
17 \( 1 - 11.2T + 289T^{2} \)
19 \( 1 + 33.1iT - 361T^{2} \)
23 \( 1 - 35.3iT - 529T^{2} \)
29 \( 1 - 35.6T + 841T^{2} \)
37 \( 1 + 11.9T + 1.36e3T^{2} \)
41 \( 1 + 45.5T + 1.68e3T^{2} \)
43 \( 1 - 16.0iT - 1.84e3T^{2} \)
47 \( 1 - 12.8iT - 2.20e3T^{2} \)
53 \( 1 + 82.8T + 2.80e3T^{2} \)
59 \( 1 + 77.3iT - 3.48e3T^{2} \)
61 \( 1 + 7.91T + 3.72e3T^{2} \)
67 \( 1 + 59.9iT - 4.48e3T^{2} \)
71 \( 1 - 63.2iT - 5.04e3T^{2} \)
73 \( 1 - 46.6T + 5.32e3T^{2} \)
79 \( 1 + 119. iT - 6.24e3T^{2} \)
83 \( 1 + 91.7iT - 6.88e3T^{2} \)
89 \( 1 + 94.4T + 7.92e3T^{2} \)
97 \( 1 + 60.3T + 9.40e3T^{2} \)
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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

−13.01938769273092700685689448865, −11.93386772774773905995115053451, −10.97499991783097955325476383163, −9.989143676282444873637644317291, −9.186481950892403588362799347738, −7.88101230944091253083501163828, −6.84606234474081022485295756594, −4.68777314350018455650446878563, −3.51061414100916784787218735020, −1.49887506481478523627884473317, 1.21024502387033502466566914020, 3.90868839073025529859738971333, 5.76007916369827729702607347346, 6.53836470504085561741775101161, 8.054385558812492182707754469092, 8.555930317336527584445966234689, 10.05572334291640711017813615729, 10.82856663145276584697838981881, 12.19407002980918492500688422499, 13.54654271322194419074808184768

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