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Properties

Label 2-136-8.5-c3-0-17
Degree $2$
Conductor $136$
Sign $-0.835 - 0.549i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
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  + (−1.85 + 2.13i)2-s + 3.39i·3-s + (−1.08 − 7.92i)4-s + 21.1i·5-s + (−7.24 − 6.32i)6-s + 36.0·7-s + (18.9 + 12.4i)8-s + 15.4·9-s + (−45.0 − 39.3i)10-s + 1.32i·11-s + (26.9 − 3.67i)12-s + 22.9i·13-s + (−66.9 + 76.7i)14-s − 71.8·15-s + (−61.6 + 17.1i)16-s + 17·17-s + ⋯
L(s)  = 1  + (−0.657 + 0.753i)2-s + 0.654i·3-s + (−0.135 − 0.990i)4-s + 1.89i·5-s + (−0.492 − 0.430i)6-s + 1.94·7-s + (0.835 + 0.549i)8-s + 0.571·9-s + (−1.42 − 1.24i)10-s + 0.0362i·11-s + (0.648 − 0.0884i)12-s + 0.490i·13-s + (−1.27 + 1.46i)14-s − 1.23·15-s + (−0.963 + 0.267i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ -0.835 - 0.549i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.418257 + 1.39655i\)
\(L(\frac12)\) \(\approx\) \(0.418257 + 1.39655i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.85 - 2.13i)T \)
17 \( 1 - 17T \)
good3 \( 1 - 3.39iT - 27T^{2} \)
5 \( 1 - 21.1iT - 125T^{2} \)
7 \( 1 - 36.0T + 343T^{2} \)
11 \( 1 - 1.32iT - 1.33e3T^{2} \)
13 \( 1 - 22.9iT - 2.19e3T^{2} \)
19 \( 1 + 50.4iT - 6.85e3T^{2} \)
23 \( 1 - 44.6T + 1.21e4T^{2} \)
29 \( 1 + 74.4iT - 2.43e4T^{2} \)
31 \( 1 + 160.T + 2.97e4T^{2} \)
37 \( 1 + 371. iT - 5.06e4T^{2} \)
41 \( 1 + 264.T + 6.89e4T^{2} \)
43 \( 1 + 144. iT - 7.95e4T^{2} \)
47 \( 1 - 445.T + 1.03e5T^{2} \)
53 \( 1 + 243. iT - 1.48e5T^{2} \)
59 \( 1 - 404. iT - 2.05e5T^{2} \)
61 \( 1 + 474. iT - 2.26e5T^{2} \)
67 \( 1 - 424. iT - 3.00e5T^{2} \)
71 \( 1 + 54.9T + 3.57e5T^{2} \)
73 \( 1 + 648.T + 3.89e5T^{2} \)
79 \( 1 + 157.T + 4.93e5T^{2} \)
83 \( 1 + 523. iT - 5.71e5T^{2} \)
89 \( 1 + 603.T + 7.04e5T^{2} \)
97 \( 1 + 142.T + 9.12e5T^{2} \)
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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

−13.81031469478220251617922807935, −11.47992285494553693140188829568, −10.87281439097051642303629381097, −10.21309508502936398147428702216, −8.953396215506401824040388453249, −7.57085979158855837487400799271, −7.03892053443302369195910140239, −5.49429509524625898471911444247, −4.18770903929809594681747148313, −1.99125933702489408154862809801, 1.08554702412379586036751554806, 1.69002648042067349636920366650, 4.30288571985932738855201771137, 5.21555505095238635788645633325, 7.57391601464746703884786315849, 8.192698726887766601198391309899, 8.948746580680261155749918445112, 10.27699861987699807593477251659, 11.56124188748251000010920466828, 12.24635606268418008511338666795

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