Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.0723 + 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.959 + 0.281i)2-s + (−0.841 + 0.540i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.654 − 0.755i)7-s + (0.654 + 0.755i)8-s + (0.415 − 0.909i)9-s + (0.415 + 0.909i)10-s + (0.654 − 0.755i)11-s − 12-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)17-s + (0.654 − 0.755i)18-s + ⋯
L(s,χ)  = 1  + (0.959 + 0.281i)2-s + (−0.841 + 0.540i)3-s + (0.841 + 0.540i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.654 − 0.755i)7-s + (0.654 + 0.755i)8-s + (0.415 − 0.909i)9-s + (0.415 + 0.909i)10-s + (0.654 − 0.755i)11-s − 12-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)17-s + (0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0723 + 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0723 + 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.0723 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (324, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ 0.0723 + 0.997i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.732082031 + 1.610944663i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.732082031 + 1.610944663i\)
\(L(\chi,1)\)  \(\approx\)  \(1.463816182 + 0.6935671234i\)
\(L(1,\chi)\)  \(\approx\)  \(1.463816182 + 0.6935671234i\)

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

−21.282847526529150291191275812907, −20.408646579204485606515851729981, −19.69776588902108531241978670752, −18.84589778394876893501325370763, −17.99852056885319091676332486614, −17.18760549358132214769145844263, −16.28368050641620228092057869286, −15.85329518868104117523004195319, −14.73087046614529655999917970688, −13.82497088375459219866685702985, −13.05283248507325661338743135605, −12.48492746165792365357570274852, −11.990577025407958764501533256124, −11.17836206712827826956442191401, −9.98494882465815785012349902425, −9.54248460793171110158226658979, −8.20039357248624887140958166517, −7.01118580625385767733243356720, −6.10080179005422199941345997808, −5.91922668419673634039209468049, −4.73775701470910177274508289613, −4.20934583622272982421814826928, −2.54422904416480816745038281892, −1.987272984900334122443171796860, −0.86110304865701288196082907005, 1.20669622236191568677302474012, 2.71601297013472756188268029510, 3.53832942696587473918541035705, 4.26505987813545959185258246413, 5.30309377133180252595011328909, 6.13648845930426037773518049610, 6.67483173801738645200538081094, 7.27085223370892972045618069689, 8.84178169758565508741496689196, 9.82556851189177947032863055175, 10.68043779547715215992872332142, 11.197159095606189691467626617176, 11.97078550152734427186516619383, 13.12419482060470537505451592604, 13.6094435640703972153588841963, 14.43227105971177907411192680562, 15.267573476313835281423833636794, 16.08950734446679284817440636450, 16.596815722063067473091016262507, 17.56107547514640687758242800606, 17.90973712556136067266745359659, 19.47912848864886827431601435399, 19.93196612165121459165169717953, 21.20452603391416663414248059226, 21.6427753243263841016149931682

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