Properties

Degree 1
Conductor 137
Sign $-0.175 - 0.984i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $-0.175 - 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (93, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ -0.175 - 0.984i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3029628221 - 0.3616026982i$
$L(\frac12,\chi)$  $\approx$  $0.3029628221 - 0.3616026982i$
$L(\chi,1)$  $\approx$  0.5162395927 - 0.1743923622i
$L(1,\chi)$  $\approx$  0.5162395927 - 0.1743923622i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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