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Properties

Label 1-1157-1157.25-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.0723 - 0.997i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (0.959 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.0723 - 0.997i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.0723 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.732082031 - 1.610944663i\)
\(L(\frac12)\) \(\approx\) \(1.732082031 - 1.610944663i\)
\(L(1)\) \(\approx\) \(1.463816182 - 0.6935671234i\)
\(L(1)\) \(\approx\) \(1.463816182 - 0.6935671234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

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

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