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Properties

Label 1-137-137.53-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.250 + 0.968i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (−0.673 − 0.739i)2-s + (0.973 − 0.228i)3-s + (−0.0922 + 0.995i)4-s + (0.0461 + 0.998i)5-s + (−0.824 − 0.565i)6-s + (−0.961 + 0.273i)7-s + (0.798 − 0.602i)8-s + (0.895 − 0.445i)9-s + (0.707 − 0.707i)10-s + (−0.995 − 0.0922i)11-s + (0.138 + 0.990i)12-s + (0.873 + 0.486i)13-s + (0.850 + 0.526i)14-s + (0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (−0.798 − 0.602i)17-s + ⋯
L(s)  = 1  + (−0.673 − 0.739i)2-s + (0.973 − 0.228i)3-s + (−0.0922 + 0.995i)4-s + (0.0461 + 0.998i)5-s + (−0.824 − 0.565i)6-s + (−0.961 + 0.273i)7-s + (0.798 − 0.602i)8-s + (0.895 − 0.445i)9-s + (0.707 − 0.707i)10-s + (−0.995 − 0.0922i)11-s + (0.138 + 0.990i)12-s + (0.873 + 0.486i)13-s + (0.850 + 0.526i)14-s + (0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (−0.798 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4542824799 + 0.5870022429i\)
\(L(\frac12)\) \(\approx\) \(0.4542824799 + 0.5870022429i\)
\(L(1)\) \(\approx\) \(0.7873250445 + 0.004402389862i\)
\(L(1)\) \(\approx\) \(0.7873250445 + 0.004402389862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.673 - 0.739i)T \)
3 \( 1 + (0.973 - 0.228i)T \)
5 \( 1 + (0.0461 + 0.998i)T \)
7 \( 1 + (-0.961 + 0.273i)T \)
11 \( 1 + (-0.995 - 0.0922i)T \)
13 \( 1 + (0.873 + 0.486i)T \)
17 \( 1 + (-0.798 - 0.602i)T \)
19 \( 1 + (-0.361 + 0.932i)T \)
23 \( 1 + (-0.824 + 0.565i)T \)
29 \( 1 + (-0.565 - 0.824i)T \)
31 \( 1 + (-0.403 + 0.914i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.914 + 0.403i)T \)
47 \( 1 + (-0.317 + 0.948i)T \)
53 \( 1 + (0.403 + 0.914i)T \)
59 \( 1 + (0.445 + 0.895i)T \)
61 \( 1 + (0.895 + 0.445i)T \)
67 \( 1 + (0.486 - 0.873i)T \)
71 \( 1 + (-0.769 - 0.638i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (0.228 - 0.973i)T \)
83 \( 1 + (0.990 + 0.138i)T \)
89 \( 1 + (0.998 - 0.0461i)T \)
97 \( 1 + (-0.638 - 0.769i)T \)
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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

−28.00889946713957195930326824343, −26.58034368738827069278392275206, −25.99021450109553766049993487671, −25.25659702585904679764728322340, −24.18423460857977233557604554346, −23.43223185564756934990582684420, −21.90130976290994487907199141190, −20.39852340285685151158060179011, −20.01725117662341114483477230400, −18.898492531262100027433701054266, −17.8150385245622421306060678430, −16.4189196500035074496800390410, −15.870336249700123985322013786945, −14.97265982737936909666853865216, −13.31577388771914559003149681040, −13.1065052327263976049866582446, −10.71557717807152944942680829086, −9.75907925037624709836447445897, −8.75943809253381210110847830821, −8.07316511284984628073791461893, −6.726542125388224192293434414455, −5.2992275179051495032802939333, −3.9363899036586611728953164715, −2.08250490262192401722967410111, −0.30016602847339579455441134186, 1.94573574155838335058671744291, 2.961258689577001773117506907219, 3.837437919600318001180300121409, 6.39945147300432930320178406631, 7.49344876242360985529341632655, 8.588679056578555458543971969444, 9.70214752886504243271886089726, 10.50459435455037412942693243513, 11.83990719678347218200075368489, 13.16150267254067897598417229558, 13.78824853847150148797911081654, 15.38390529110930402571511728000, 16.20092610738799856474706371825, 18.01305907843425052777083769113, 18.63819252904063098576685816639, 19.27297133008182165880419766476, 20.37214922790179010956295422044, 21.30624221379593553970646631321, 22.26422738474747556039501593490, 23.43494308737948893089213788803, 25.12509933467122080822143647878, 25.91227950518261305071677425217, 26.352431527182169464864231576085, 27.34567623659245601498297070512, 28.80347207159729423169447059045

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