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Properties

Label 1-1157-1157.275-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.0633 + 0.997i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (0.371 − 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (0.909 − 0.415i)5-s + (−0.371 − 0.928i)6-s + (0.814 + 0.580i)7-s + (−0.909 + 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (−0.814 + 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (0.371 − 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (−0.909 − 0.415i)18-s + ⋯
L(s)  = 1  + (0.371 − 0.928i)2-s + (0.723 − 0.690i)3-s + (−0.723 − 0.690i)4-s + (0.909 − 0.415i)5-s + (−0.371 − 0.928i)6-s + (0.814 + 0.580i)7-s + (−0.909 + 0.415i)8-s + (0.0475 − 0.998i)9-s + (−0.0475 − 0.998i)10-s + (−0.814 + 0.580i)11-s − 12-s + (0.841 − 0.540i)14-s + (0.371 − 0.928i)15-s + (0.0475 + 0.998i)16-s + (−0.928 + 0.371i)17-s + (−0.909 − 0.415i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8098538010 - 0.8628506072i\)
\(L(\frac12)\) \(\approx\) \(-0.8098538010 - 0.8628506072i\)
\(L(1)\) \(\approx\) \(0.9404812293 - 1.050620986i\)
\(L(1)\) \(\approx\) \(0.9404812293 - 1.050620986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.371 - 0.928i)T \)
3 \( 1 + (0.723 - 0.690i)T \)
5 \( 1 + (0.909 - 0.415i)T \)
7 \( 1 + (0.814 + 0.580i)T \)
11 \( 1 + (-0.814 + 0.580i)T \)
17 \( 1 + (-0.928 + 0.371i)T \)
19 \( 1 + (-0.998 - 0.0475i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (-0.995 - 0.0950i)T \)
31 \( 1 + (-0.540 - 0.841i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.971 - 0.235i)T \)
43 \( 1 + (0.995 - 0.0950i)T \)
47 \( 1 + (-0.281 - 0.959i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (-0.690 + 0.723i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
67 \( 1 + (-0.690 - 0.723i)T \)
71 \( 1 + (0.0950 + 0.995i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (-0.814 - 0.580i)T \)
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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

−21.70514197982742621437857399254, −20.95045936432742176436826562870, −20.59870088531944450309225318369, −19.21573993296690555456406748809, −18.4296326576384815275893112628, −17.554359192369313298146102688547, −16.98431966385150449544418927690, −16.14089488667534274933698450023, −15.242769547674970737530123399159, −14.750493964375360732768209231977, −13.86023729748498939119297486756, −13.55847456222367431540955550397, −12.743225381435428201845744178103, −11.00813965561790765732674896457, −10.71045493604075541984886815674, −9.49914529650697739305635143668, −8.86610431789281802535315706612, −8.03721984728705963590487685432, −7.26240542887853959402492649568, −6.33427089000491298754971580138, −5.17548551513375032433749312282, −4.8131774029203773141241820604, −3.64637865637360564660848213449, −2.840183990759270756116956681677, −1.7439248858273547169254392071, 0.15615980992515755517046124468, 1.5264096473653236081035113214, 2.12039937805066335984907145753, 2.608612952473369449171589053303, 3.998991268168389198937716770151, 4.920976019475871733198560558138, 5.69654941245345120405606031712, 6.66322599479634586182509808834, 7.91804407647688162136340791548, 8.86880004078889056384218414219, 9.145757387398304766025229610642, 10.319726068184546173496982492196, 11.04368071530614379536035138833, 12.13404264126602366381390728242, 12.864474705650048109651783866216, 13.20768908519324137956117414592, 14.07837976749692450427594859352, 14.94051212442278841307532190465, 15.29677754269073845085506521879, 17.07834116185135662684327095055, 17.735014911315618103101786522750, 18.36774723915814532312122481243, 18.95463008734960297027971647969, 19.922490999002619597359914186633, 20.79296758450926618730277979201

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