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Properties

Label 2-124-124.123-c3-0-9
Degree $2$
Conductor $124$
Sign $0.701 - 0.712i$
Analytic cond. $7.31623$
Root an. cond. $2.70485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.09 − 1.89i)2-s − 10.0·3-s + (0.807 − 7.95i)4-s − 5.59·5-s + (−21.1 + 19.1i)6-s + 15.9i·7-s + (−13.4 − 18.2i)8-s + 74.5·9-s + (−11.7 + 10.6i)10-s + 22.3·11-s + (−8.13 + 80.2i)12-s + 62.3i·13-s + (30.2 + 33.4i)14-s + 56.4·15-s + (−62.6 − 12.8i)16-s − 61.2i·17-s + ⋯
L(s)  = 1  + (0.741 − 0.670i)2-s − 1.93·3-s + (0.100 − 0.994i)4-s − 0.500·5-s + (−1.43 + 1.30i)6-s + 0.860i·7-s + (−0.592 − 0.805i)8-s + 2.76·9-s + (−0.371 + 0.335i)10-s + 0.613·11-s + (−0.195 + 1.92i)12-s + 1.32i·13-s + (0.576 + 0.638i)14-s + 0.971·15-s + (−0.979 − 0.200i)16-s − 0.874i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.736687 + 0.308349i\)
\(L(\frac12)\) \(\approx\) \(0.736687 + 0.308349i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.09 + 1.89i)T \)
31 \( 1 + (-134. - 108. i)T \)
good3 \( 1 + 10.0T + 27T^{2} \)
5 \( 1 + 5.59T + 125T^{2} \)
7 \( 1 - 15.9iT - 343T^{2} \)
11 \( 1 - 22.3T + 1.33e3T^{2} \)
13 \( 1 - 62.3iT - 2.19e3T^{2} \)
17 \( 1 + 61.2iT - 4.91e3T^{2} \)
19 \( 1 - 137. iT - 6.85e3T^{2} \)
23 \( 1 - 30.7T + 1.21e4T^{2} \)
29 \( 1 - 211. iT - 2.43e4T^{2} \)
37 \( 1 + 115. iT - 5.06e4T^{2} \)
41 \( 1 + 15.6T + 6.89e4T^{2} \)
43 \( 1 + 280.T + 7.95e4T^{2} \)
47 \( 1 + 1.51iT - 1.03e5T^{2} \)
53 \( 1 - 214. iT - 1.48e5T^{2} \)
59 \( 1 - 494. iT - 2.05e5T^{2} \)
61 \( 1 - 359. iT - 2.26e5T^{2} \)
67 \( 1 + 452. iT - 3.00e5T^{2} \)
71 \( 1 + 152. iT - 3.57e5T^{2} \)
73 \( 1 + 45.8iT - 3.89e5T^{2} \)
79 \( 1 - 42.6T + 4.93e5T^{2} \)
83 \( 1 - 320.T + 5.71e5T^{2} \)
89 \( 1 - 859. iT - 7.04e5T^{2} \)
97 \( 1 + 763.T + 9.12e5T^{2} \)
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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

−12.38531713101871918829333853166, −11.99620177567562470596073630664, −11.45341470221453178758679230246, −10.39184192162346936614078693993, −9.297764073666190596049281663452, −6.99089416128416193203909728912, −6.06830718671395296885879323919, −5.08396725749325030259120041932, −3.99972433656409084291367610888, −1.48424785450540712629966299541, 0.44854312131684021706410617364, 3.94442127997562356662105305421, 4.89482121167070066663525110965, 6.06747683086956111937884053155, 6.91316314179098697826561242097, 7.938239411202563965477173078163, 10.03326002008883298701101100375, 11.16950228988133115707168209006, 11.73512545545243312814991566614, 12.83354572273132465942523537240

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