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Properties

Label 2-124-124.15-c1-0-0
Degree $2$
Conductor $124$
Sign $-0.323 - 0.946i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
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  + (−1.39 − 0.221i)2-s + (−2.48 − 1.80i)3-s + (1.90 + 0.618i)4-s − 1.61·5-s + (3.07 + 3.07i)6-s + (0.224 + 0.0729i)7-s + (−2.52 − 1.28i)8-s + (1.99 + 6.15i)9-s + (2.26 + 0.357i)10-s + (−1.90 + 5.85i)11-s + (−3.61 − 4.97i)12-s + (−0.263 + 0.363i)13-s + (−0.297 − 0.151i)14-s + (4.02 + 2.92i)15-s + (3.23 + 2.35i)16-s + (−1.54 + 0.502i)17-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)2-s + (−1.43 − 1.04i)3-s + (0.951 + 0.309i)4-s − 0.723·5-s + (1.25 + 1.25i)6-s + (0.0848 + 0.0275i)7-s + (−0.891 − 0.453i)8-s + (0.666 + 2.05i)9-s + (0.714 + 0.113i)10-s + (−0.573 + 1.76i)11-s + (−1.04 − 1.43i)12-s + (−0.0732 + 0.100i)13-s + (−0.0795 − 0.0405i)14-s + (1.04 + 0.755i)15-s + (0.809 + 0.587i)16-s + (−0.374 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0563782 + 0.0788195i\)
\(L(\frac12)\) \(\approx\) \(0.0563782 + 0.0788195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.221i)T \)
31 \( 1 + (1.76 - 5.28i)T \)
good3 \( 1 + (2.48 + 1.80i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
7 \( 1 + (-0.224 - 0.0729i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.90 - 5.85i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.263 - 0.363i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.54 - 0.502i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.93 + 4.04i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.224 - 0.690i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.30 + 5.93i)T + (-8.96 + 27.5i)T^{2} \)
37 \( 1 - 9.23iT - 37T^{2} \)
41 \( 1 + (-4.23 + 3.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.12 - 1.54i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (3.30 - 4.54i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.78 - 2.85i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.21 + 4.42i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 - 0.898iT - 61T^{2} \)
67 \( 1 + 5.76iT - 67T^{2} \)
71 \( 1 + (-6.29 + 2.04i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.80 + 0.587i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.07 - 6.38i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.34 + 3.88i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.33 + 0.759i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.454 + 1.40i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.07259498406830053352695974432, −12.40825562727354198420379350493, −11.58287834383934598695012932900, −10.87832866423630376061455977114, −9.737265820176241624672846405988, −8.016219053947058930015526912100, −7.25412453127035627731006535825, −6.42580932404463605550462538552, −4.78447691672486097476754939985, −1.96446641156117055601860268381, 0.15855955388960393802447133357, 3.70293624395233719707274430253, 5.43440821100125345400309393070, 6.26383564222240616521385959331, 7.81676856566750777767254676717, 8.983846161139569866985002760426, 10.20010310353925893445491262629, 11.13554602148151672786999941741, 11.33765907431526804935401746670, 12.64028469514367913192375147902

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