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Properties

Label 2-136-136.13-c1-0-7
Degree $2$
Conductor $136$
Sign $0.460 + 0.887i$
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  + (−1.17 − 0.780i)2-s + (0.662 − 0.662i)3-s + (0.780 + 1.84i)4-s + (−1.29 + 0.263i)6-s + (2.56 − 2.56i)7-s + (0.516 − 2.78i)8-s + 2.12i·9-s + (0.662 + 0.662i)11-s + (1.73 + 0.702i)12-s − 6.04i·13-s + (−5.02 + 1.02i)14-s + (−2.78 + 2.87i)16-s + (−4 − i)17-s + (1.65 − 2.50i)18-s + 4.71·19-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (0.382 − 0.382i)3-s + (0.390 + 0.920i)4-s + (−0.529 + 0.107i)6-s + (0.968 − 0.968i)7-s + (0.182 − 0.983i)8-s + 0.707i·9-s + (0.199 + 0.199i)11-s + (0.501 + 0.202i)12-s − 1.67i·13-s + (−1.34 + 0.272i)14-s + (−0.695 + 0.718i)16-s + (−0.970 − 0.242i)17-s + (0.390 − 0.590i)18-s + 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775663 - 0.471148i\)
\(L(\frac12)\) \(\approx\) \(0.775663 - 0.471148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.780i)T \)
17 \( 1 + (4 + i)T \)
good3 \( 1 + (-0.662 + 0.662i)T - 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (-2.56 + 2.56i)T - 7iT^{2} \)
11 \( 1 + (-0.662 - 0.662i)T + 11iT^{2} \)
13 \( 1 + 6.04iT - 13T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 + (1.43 - 1.43i)T - 23iT^{2} \)
29 \( 1 + (3.39 - 3.39i)T - 29iT^{2} \)
31 \( 1 + (-2.56 - 2.56i)T + 31iT^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 37iT^{2} \)
41 \( 1 + (2.12 - 2.12i)T - 41iT^{2} \)
43 \( 1 + 6.04T + 43T^{2} \)
47 \( 1 - 2.87T + 47T^{2} \)
53 \( 1 - 2.64T + 53T^{2} \)
59 \( 1 - 6.04T + 59T^{2} \)
61 \( 1 + (2.64 + 2.64i)T + 61iT^{2} \)
67 \( 1 - 6.04iT - 67T^{2} \)
71 \( 1 + (11.6 + 11.6i)T + 71iT^{2} \)
73 \( 1 + (6.12 + 6.12i)T + 73iT^{2} \)
79 \( 1 + (-3.68 + 3.68i)T - 79iT^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−13.15083109100705327191311275365, −11.79085928703580431020179518270, −10.84706073377078809647324174403, −10.14632692298437516526527030600, −8.708206274529497281561497060364, −7.77329443863889448304014133846, −7.19006186935255852324471472202, −4.97697693800751742145711146028, −3.23311753337854153139593858555, −1.50473674719255440638488070380, 2.08704047584068127083786178011, 4.40277848690367037640259826943, 5.87674621948186286833855982824, 7.01178530982199301174287921502, 8.516614094530276336749722113726, 8.959921223256955098903496238530, 9.949620524805587241871187848198, 11.42126473032002471777788249534, 11.90608767328638769715741087698, 13.92204816506479457875963358449

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