Properties

Label 2-138-69.68-c3-0-22
Degree $2$
Conductor $138$
Sign $-0.901 - 0.431i$
Analytic cond. $8.14226$
Root an. cond. $2.85346$
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  − 2i·2-s + (−1.34 − 5.01i)3-s − 4·4-s + 4.58·5-s + (−10.0 + 2.69i)6-s − 31.2i·7-s + 8i·8-s + (−23.3 + 13.5i)9-s − 9.17i·10-s − 0.810·11-s + (5.38 + 20.0i)12-s − 14.3·13-s − 62.5·14-s + (−6.18 − 23.0i)15-s + 16·16-s − 45.4·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.259 − 0.965i)3-s − 0.5·4-s + 0.410·5-s + (−0.682 + 0.183i)6-s − 1.68i·7-s + 0.353i·8-s + (−0.865 + 0.500i)9-s − 0.290i·10-s − 0.0222·11-s + (0.129 + 0.482i)12-s − 0.306·13-s − 1.19·14-s + (−0.106 − 0.396i)15-s + 0.250·16-s − 0.647·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 - 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.901 - 0.431i$
Analytic conductor: \(8.14226\)
Root analytic conductor: \(2.85346\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :3/2),\ -0.901 - 0.431i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.228229 + 1.00519i\)
\(L(\frac12)\) \(\approx\) \(0.228229 + 1.00519i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + (1.34 + 5.01i)T \)
23 \( 1 + (-83.7 - 71.7i)T \)
good5 \( 1 - 4.58T + 125T^{2} \)
7 \( 1 + 31.2iT - 343T^{2} \)
11 \( 1 + 0.810T + 1.33e3T^{2} \)
13 \( 1 + 14.3T + 2.19e3T^{2} \)
17 \( 1 + 45.4T + 4.91e3T^{2} \)
19 \( 1 - 11.6iT - 6.85e3T^{2} \)
29 \( 1 - 98.1iT - 2.43e4T^{2} \)
31 \( 1 + 56.6T + 2.97e4T^{2} \)
37 \( 1 + 398. iT - 5.06e4T^{2} \)
41 \( 1 + 97.5iT - 6.89e4T^{2} \)
43 \( 1 + 323. iT - 7.95e4T^{2} \)
47 \( 1 + 341. iT - 1.03e5T^{2} \)
53 \( 1 - 500.T + 1.48e5T^{2} \)
59 \( 1 + 37.2iT - 2.05e5T^{2} \)
61 \( 1 + 667. iT - 2.26e5T^{2} \)
67 \( 1 - 791. iT - 3.00e5T^{2} \)
71 \( 1 + 854. iT - 3.57e5T^{2} \)
73 \( 1 - 119.T + 3.89e5T^{2} \)
79 \( 1 + 604. iT - 4.93e5T^{2} \)
83 \( 1 - 813.T + 5.71e5T^{2} \)
89 \( 1 - 460.T + 7.04e5T^{2} \)
97 \( 1 - 710. iT - 9.12e5T^{2} \)
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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

−12.18080589516278824073094711073, −11.05526211175798328993806380130, −10.40589180313961280760121727224, −9.099870189852099897043912916051, −7.65903696418976344724100150272, −6.85745320600792328038869115664, −5.32636323578458858261980205908, −3.76487926029836402177312407364, −1.95255233149210696757821774786, −0.52357550813331929175221369445, 2.71765389923765658609212326108, 4.59770596060659905144187822380, 5.58776914869556374680367971498, 6.43164050333151676400607274580, 8.296918897425750121741578012918, 9.140528435628506244014309076939, 9.885408585249066245400892727445, 11.27771482199113960646701151014, 12.22266340815279296583964347128, 13.41656084599584153079269457166

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