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Properties

Label 2-136-136.101-c1-0-5
Degree $2$
Conductor $136$
Sign $0.772 + 0.634i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
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  + (−0.632 + 1.26i)2-s − 2.88·3-s + (−1.20 − 1.59i)4-s + 0.914·5-s + (1.82 − 3.64i)6-s − 2.61i·7-s + (2.78 − 0.507i)8-s + 5.30·9-s + (−0.578 + 1.15i)10-s + 0.394·11-s + (3.45 + 4.60i)12-s − 3.33i·13-s + (3.31 + 1.65i)14-s − 2.63·15-s + (−1.11 + 3.84i)16-s + (−2.66 − 3.14i)17-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)2-s − 1.66·3-s + (−0.600 − 0.799i)4-s + 0.408·5-s + (0.743 − 1.48i)6-s − 0.990i·7-s + (0.983 − 0.179i)8-s + 1.76·9-s + (−0.182 + 0.365i)10-s + 0.118·11-s + (0.998 + 1.33i)12-s − 0.924i·13-s + (0.885 + 0.442i)14-s − 0.680·15-s + (−0.279 + 0.960i)16-s + (−0.646 − 0.762i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.772 + 0.634i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.772 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417346 - 0.149341i\)
\(L(\frac12)\) \(\approx\) \(0.417346 - 0.149341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.632 - 1.26i)T \)
17 \( 1 + (2.66 + 3.14i)T \)
good3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 - 0.914T + 5T^{2} \)
7 \( 1 + 2.61iT - 7T^{2} \)
11 \( 1 - 0.394T + 11T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
19 \( 1 + 6.87iT - 19T^{2} \)
23 \( 1 + 0.692iT - 23T^{2} \)
29 \( 1 - 8.20T + 29T^{2} \)
31 \( 1 - 7.59iT - 31T^{2} \)
37 \( 1 + 8.98T + 37T^{2} \)
41 \( 1 + 6.90iT - 41T^{2} \)
43 \( 1 + 2.18iT - 43T^{2} \)
47 \( 1 - 5.96T + 47T^{2} \)
53 \( 1 - 7.24iT - 53T^{2} \)
59 \( 1 - 0.845iT - 59T^{2} \)
61 \( 1 - 3.22T + 61T^{2} \)
67 \( 1 - 8.21iT - 67T^{2} \)
71 \( 1 - 8.91iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 13.8iT - 79T^{2} \)
83 \( 1 + 5.82iT - 83T^{2} \)
89 \( 1 - 2.13T + 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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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

−13.28577663574979219176177663268, −11.97517441459197775483659475043, −10.62142132062014193534524449755, −10.38907070705963447585892014352, −8.943296453071143843627684500020, −7.25779183710072215357007285210, −6.65126518706256397279807693297, −5.44888198043589564923281364194, −4.59151954454732076201055312257, −0.66876644394539057635847006229, 1.84834763315340779652026627356, 4.20548771166596522176090034819, 5.55790968717659744259914537962, 6.54620505358193082293968194360, 8.278889250382490786540418130473, 9.573361758217985598211130352806, 10.39507268270566019088454679428, 11.45481148667285467660406522016, 12.02783147211167624467962662527, 12.74887156169085026279294663007

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