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Properties

Label 2-124-124.123-c1-0-4
Degree $2$
Conductor $124$
Sign $0.898 + 0.439i$
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 + i)2-s − 2.44·3-s − 2i·4-s + 5-s + (2.44 − 2.44i)6-s − 3i·7-s + (2 + 2i)8-s + 2.99·9-s + (−1 + i)10-s + 4.89·11-s + 4.89i·12-s − 2.44i·13-s + (3 + 3i)14-s − 2.44·15-s − 4·16-s − 7.34i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.41·3-s − i·4-s + 0.447·5-s + (0.999 − 0.999i)6-s − 1.13i·7-s + (0.707 + 0.707i)8-s + 0.999·9-s + (−0.316 + 0.316i)10-s + 1.47·11-s + 1.41i·12-s − 0.679i·13-s + (0.801 + 0.801i)14-s − 0.632·15-s − 16-s − 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.898 + 0.439i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506431 - 0.117385i\)
\(L(\frac12)\) \(\approx\) \(0.506431 - 0.117385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
31 \( 1 + (-2.44 + 5i)T \)
good3 \( 1 + 2.44T + 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 2.44iT - 29T^{2} \)
37 \( 1 - 4.89iT - 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 9.79iT - 53T^{2} \)
59 \( 1 - 11iT - 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 9.79T + 83T^{2} \)
89 \( 1 - 2.44iT - 89T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−13.60088581940385376127439798442, −11.95387882755326720534499084337, −11.13974324170909440512295227739, −10.15654368015727119821522327341, −9.353349697608918861458412722843, −7.61813312078094077886396957695, −6.65246070053453051526810544784, −5.79093213079397994464081785382, −4.52027768891085925261786286561, −0.907116750003131124119676193492, 1.78788613401014813332776415773, 4.11195671105746595029527799726, 5.83978548326827659751706140955, 6.65472337270925702087526487951, 8.499443123739439181716666084497, 9.446408232563317604710590508309, 10.50198921180952591009804651330, 11.61046402586476651250067449013, 11.99212441649842335878108995467, 12.92124273433719718641635358670

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