Properties

Label 2-124-124.123-c3-0-12
Degree $2$
Conductor $124$
Sign $-0.655 - 0.754i$
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  + (−1.63 + 2.30i)2-s + 0.254·3-s + (−2.63 − 7.55i)4-s + 3.31·5-s + (−0.416 + 0.586i)6-s + 7.54i·7-s + (21.7 + 6.29i)8-s − 26.9·9-s + (−5.42 + 7.63i)10-s + 43.9·11-s + (−0.670 − 1.92i)12-s + 69.9i·13-s + (−17.4 − 12.3i)14-s + 0.842·15-s + (−50.1 + 39.8i)16-s + 85.3i·17-s + ⋯
L(s)  = 1  + (−0.579 + 0.815i)2-s + 0.0489·3-s + (−0.329 − 0.944i)4-s + 0.296·5-s + (−0.0283 + 0.0399i)6-s + 0.407i·7-s + (0.960 + 0.278i)8-s − 0.997·9-s + (−0.171 + 0.241i)10-s + 1.20·11-s + (−0.0161 − 0.0462i)12-s + 1.49i·13-s + (−0.332 − 0.235i)14-s + 0.0144·15-s + (−0.782 + 0.622i)16-s + 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.655 - 0.754i$
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: \((2,\ 124,\ (\ :3/2),\ -0.655 - 0.754i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.406969 + 0.892764i\)
\(L(\frac12)\) \(\approx\) \(0.406969 + 0.892764i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.63 - 2.30i)T \)
31 \( 1 + (-160. + 63.9i)T \)
good3 \( 1 - 0.254T + 27T^{2} \)
5 \( 1 - 3.31T + 125T^{2} \)
7 \( 1 - 7.54iT - 343T^{2} \)
11 \( 1 - 43.9T + 1.33e3T^{2} \)
13 \( 1 - 69.9iT - 2.19e3T^{2} \)
17 \( 1 - 85.3iT - 4.91e3T^{2} \)
19 \( 1 - 8.51iT - 6.85e3T^{2} \)
23 \( 1 + 129.T + 1.21e4T^{2} \)
29 \( 1 - 100. iT - 2.43e4T^{2} \)
37 \( 1 - 297. iT - 5.06e4T^{2} \)
41 \( 1 + 134.T + 6.89e4T^{2} \)
43 \( 1 - 495.T + 7.95e4T^{2} \)
47 \( 1 - 440. iT - 1.03e5T^{2} \)
53 \( 1 + 174. iT - 1.48e5T^{2} \)
59 \( 1 + 365. iT - 2.05e5T^{2} \)
61 \( 1 + 608. iT - 2.26e5T^{2} \)
67 \( 1 - 790. iT - 3.00e5T^{2} \)
71 \( 1 + 586. iT - 3.57e5T^{2} \)
73 \( 1 + 771. iT - 3.89e5T^{2} \)
79 \( 1 - 307.T + 4.93e5T^{2} \)
83 \( 1 + 862.T + 5.71e5T^{2} \)
89 \( 1 - 413. iT - 7.04e5T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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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

−13.86772493578106908434350092599, −12.11773211919731395240275992084, −11.19165327074809616657045156886, −9.788968471292548969218624221956, −8.979861546895877445636752513309, −8.110828084589863678716288045879, −6.51817906529500696525553730664, −5.91085010279860606548576832283, −4.21727937179859781131003291067, −1.78420037450274067565660485715, 0.62853919980808708191609435574, 2.58239855317200368873755239346, 3.95872301233439474180995433820, 5.73876794362405442969408096791, 7.38495783390427935802858904788, 8.477513562508930221516748475898, 9.505950140873069196911661067996, 10.42487425591662448284345472263, 11.53915963014322346482169018726, 12.23567696150183254981342197597

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