Properties

Label 1-137-137.68-r0-0-0
Degree $1$
Conductor $137$
Sign $0.405 + 0.914i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0922 + 0.995i)2-s + (−0.895 + 0.445i)3-s + (−0.982 − 0.183i)4-s + (0.995 − 0.0922i)5-s + (−0.361 − 0.932i)6-s + (0.850 − 0.526i)7-s + (0.273 − 0.961i)8-s + (0.602 − 0.798i)9-s + i·10-s + (0.982 + 0.183i)11-s + (0.961 − 0.273i)12-s + (−0.526 − 0.850i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 + 0.361i)16-s + (0.273 + 0.961i)17-s + ⋯
L(s)  = 1  + (−0.0922 + 0.995i)2-s + (−0.895 + 0.445i)3-s + (−0.982 − 0.183i)4-s + (0.995 − 0.0922i)5-s + (−0.361 − 0.932i)6-s + (0.850 − 0.526i)7-s + (0.273 − 0.961i)8-s + (0.602 − 0.798i)9-s + i·10-s + (0.982 + 0.183i)11-s + (0.961 − 0.273i)12-s + (−0.526 − 0.850i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 + 0.361i)16-s + (0.273 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.405 + 0.914i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.405 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7808762099 + 0.5078423354i\)
\(L(\frac12)\) \(\approx\) \(0.7808762099 + 0.5078423354i\)
\(L(1)\) \(\approx\) \(0.8185522859 + 0.4165507252i\)
\(L(1)\) \(\approx\) \(0.8185522859 + 0.4165507252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.0922 + 0.995i)T \)
3 \( 1 + (-0.895 + 0.445i)T \)
5 \( 1 + (0.995 - 0.0922i)T \)
7 \( 1 + (0.850 - 0.526i)T \)
11 \( 1 + (0.982 + 0.183i)T \)
13 \( 1 + (-0.526 - 0.850i)T \)
17 \( 1 + (0.273 + 0.961i)T \)
19 \( 1 + (-0.739 - 0.673i)T \)
23 \( 1 + (-0.361 + 0.932i)T \)
29 \( 1 + (0.361 - 0.932i)T \)
31 \( 1 + (0.673 + 0.739i)T \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 + (-0.673 + 0.739i)T \)
47 \( 1 + (0.798 + 0.602i)T \)
53 \( 1 + (0.673 - 0.739i)T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (0.602 + 0.798i)T \)
67 \( 1 + (0.526 + 0.850i)T \)
71 \( 1 + (-0.183 - 0.982i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.895 + 0.445i)T \)
83 \( 1 + (-0.961 - 0.273i)T \)
89 \( 1 + (-0.995 + 0.0922i)T \)
97 \( 1 + (0.183 - 0.982i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.39883431441514570612956976731, −27.69147772984674460963825463327, −26.71063482490298717676665999231, −25.1361532054648815330135602149, −24.33857611973092564659411287342, −23.062933000269711785186065367927, −21.99365898671804950707522875629, −21.56625022884445528373508362219, −20.435888405370029955839158578014, −18.89212217322280605549606849193, −18.38378703628481327861827849860, −17.31244716870102874363941781624, −16.71772349521526142613951440691, −14.47648776939408871189382428505, −13.79712182732714277090277623608, −12.377138525765500846140754844509, −11.780300220318825066999048358995, −10.709772207678511779088437054880, −9.62934728668376446370866457305, −8.46314183911226494978334613050, −6.73492545420330265449021206015, −5.45839066821800702244350827421, −4.46041982337806630026002731566, −2.33237143805120061890867594741, −1.41321056218369874846945569172, 1.28749176158212489810045157377, 4.08262252620497943840729333037, 5.11412117214171061736339680914, 6.05185271915462683392988037805, 7.09721640631181243540450672709, 8.58908918109575516289388170355, 9.84429328003489414847153678299, 10.589881756064400788051726588079, 12.18990132507522750528025747365, 13.4391645776762580425750416645, 14.552894673349332017257550697, 15.39366097009241265310537711630, 16.83221454847208103606977760312, 17.48973779061305127154996098338, 17.72208490967433105292841496253, 19.46156566537463638182710739151, 21.08511699774524207947774610383, 21.87397918374482529627782984413, 22.7686135231213944212909167837, 23.82639909976898173909407742269, 24.62119041127200400909364595640, 25.653653480787573579536188326559, 26.742309190059350544907128261329, 27.651563656060032166583632102060, 28.25113654601758556108525812101

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