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Properties

Label 1-137-137.28-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.175 + 0.984i$
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  + (−0.932 − 0.361i)2-s + (−0.961 + 0.273i)3-s + (0.739 + 0.673i)4-s + (0.361 + 0.932i)5-s + (0.995 + 0.0922i)6-s + (0.602 + 0.798i)7-s + (−0.445 − 0.895i)8-s + (0.850 − 0.526i)9-s − i·10-s + (−0.739 − 0.673i)11-s + (−0.895 − 0.445i)12-s + (−0.798 + 0.602i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.445 + 0.895i)17-s + ⋯
L(s)  = 1  + (−0.932 − 0.361i)2-s + (−0.961 + 0.273i)3-s + (0.739 + 0.673i)4-s + (0.361 + 0.932i)5-s + (0.995 + 0.0922i)6-s + (0.602 + 0.798i)7-s + (−0.445 − 0.895i)8-s + (0.850 − 0.526i)9-s − i·10-s + (−0.739 − 0.673i)11-s + (−0.895 − 0.445i)12-s + (−0.798 + 0.602i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.445 + 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.175 + 0.984i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.175 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3029628221 + 0.3616026982i\)
\(L(\frac12)\) \(\approx\) \(0.3029628221 + 0.3616026982i\)
\(L(1)\) \(\approx\) \(0.5162395927 + 0.1743923622i\)
\(L(1)\) \(\approx\) \(0.5162395927 + 0.1743923622i\)

Euler product

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

−28.074332356278157637409313314177, −27.42822246672731931982748253741, −26.44050510770377572268559768814, −24.98965933702741110486508543296, −24.38387801696804802776047197399, −23.57674921218163825114010377594, −22.47132430026593416371038053327, −20.73255257702534206365295872527, −20.341053852503708023315917267838, −18.82017017507493761434925393181, −17.65561244309487979311414502709, −17.325446122713478316183612144777, −16.338763497486807169835960605207, −15.33805500615665035672614277763, −13.72202153988547259468002182904, −12.515803281632800377121175276487, −11.35961710562221032761956071119, −10.31206264857840386609623236408, −9.387803576129033231748972104922, −7.74479636178076107117979036714, −7.17350265114876119809867830944, −5.46949438515665272497551948343, −4.87279561504745163323213113816, −1.98203957993828235380231900346, −0.63818827098430874098999969609, 1.80662935211530490054504601320, 3.19805811971706033875483453054, 5.16988775719197367255700045389, 6.3961888933527439891871972001, 7.48030801644652975227584069026, 8.973717614787029344203420196047, 10.09987041577740670079138115063, 11.048598951399068916975538214590, 11.65076783226914161470422116572, 12.93672828917108168808200520669, 14.76159615072489965548954721728, 15.71946672597357478430895616797, 16.855368480469909089891392959188, 17.84340416669854411442091250741, 18.4278509691287170352795354410, 19.34294938808230617884824303550, 21.11816321332142108413514775548, 21.5934739737576754740636043884, 22.42120125417514214705924984353, 23.98107512594234759428513225628, 24.82282026236668232720693253827, 26.34586990591146510956650737924, 26.737453395019118536049543866636, 27.83585866733469767017624426955, 28.8278621686581711387915531900

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