Properties

Label 1-107-107.13-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.714 + 0.699i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
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  + (0.674 + 0.737i)2-s + (0.0296 + 0.999i)3-s + (−0.0887 + 0.996i)4-s + (0.263 + 0.964i)5-s + (−0.717 + 0.696i)6-s + (−0.430 − 0.902i)7-s + (−0.794 + 0.606i)8-s + (−0.998 + 0.0592i)9-s + (−0.533 + 0.845i)10-s + (0.829 − 0.558i)11-s + (−0.998 − 0.0592i)12-s + (0.582 − 0.812i)13-s + (0.375 − 0.926i)14-s + (−0.956 + 0.292i)15-s + (−0.984 − 0.176i)16-s + (0.482 − 0.875i)17-s + ⋯
L(s)  = 1  + (0.674 + 0.737i)2-s + (0.0296 + 0.999i)3-s + (−0.0887 + 0.996i)4-s + (0.263 + 0.964i)5-s + (−0.717 + 0.696i)6-s + (−0.430 − 0.902i)7-s + (−0.794 + 0.606i)8-s + (−0.998 + 0.0592i)9-s + (−0.533 + 0.845i)10-s + (0.829 − 0.558i)11-s + (−0.998 − 0.0592i)12-s + (0.582 − 0.812i)13-s + (0.375 − 0.926i)14-s + (−0.956 + 0.292i)15-s + (−0.984 − 0.176i)16-s + (0.482 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5292259938 + 1.296022750i\)
\(L(\frac12)\) \(\approx\) \(0.5292259938 + 1.296022750i\)
\(L(1)\) \(\approx\) \(0.9479843107 + 1.013534740i\)
\(L(1)\) \(\approx\) \(0.9479843107 + 1.013534740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.674 + 0.737i)T \)
3 \( 1 + (0.0296 + 0.999i)T \)
5 \( 1 + (0.263 + 0.964i)T \)
7 \( 1 + (-0.430 - 0.902i)T \)
11 \( 1 + (0.829 - 0.558i)T \)
13 \( 1 + (0.582 - 0.812i)T \)
17 \( 1 + (0.482 - 0.875i)T \)
19 \( 1 + (-0.320 + 0.947i)T \)
23 \( 1 + (0.375 + 0.926i)T \)
29 \( 1 + (0.147 + 0.989i)T \)
31 \( 1 + (-0.915 - 0.403i)T \)
37 \( 1 + (0.992 + 0.118i)T \)
41 \( 1 + (-0.205 + 0.978i)T \)
43 \( 1 + (0.263 - 0.964i)T \)
47 \( 1 + (-0.205 - 0.978i)T \)
53 \( 1 + (0.674 - 0.737i)T \)
59 \( 1 + (0.147 - 0.989i)T \)
61 \( 1 + (-0.430 + 0.902i)T \)
67 \( 1 + (-0.794 - 0.606i)T \)
71 \( 1 + (0.0296 - 0.999i)T \)
73 \( 1 + (0.972 - 0.234i)T \)
79 \( 1 + (0.937 - 0.348i)T \)
83 \( 1 + (-0.630 - 0.776i)T \)
89 \( 1 + (-0.717 - 0.696i)T \)
97 \( 1 + (-0.533 + 0.845i)T \)
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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

−29.162605986735645127847482468914, −28.459122286674523060025348184604, −27.91409918641063381747586987042, −25.75450677400696503535886748768, −24.84958786734971775509619549835, −24.05037154905270037142975567383, −23.10631539605663585464164646871, −21.94089397749299465919061056915, −20.93416503973430557292346035059, −19.76339016344628514749835613448, −19.102765954486176309501715859488, −17.95382610802355063182527450912, −16.65575828563143374483010802081, −15.10570873579943192884147790190, −13.95819979492213388274862135904, −12.82112067034759824190970641028, −12.355385483656549918284161788, −11.29572463583725440255090771400, −9.43247318205093175840973625384, −8.65731795296166500450205438487, −6.58573813601668443642991663103, −5.72404664016205691798423736556, −4.23947488172332845425840642442, −2.45377046004863501450960761074, −1.33943407835253767921008958075, 3.23409950137990950903661357763, 3.75988824555213078590662321126, 5.44885584133015591772510092973, 6.483256701307405428368709891326, 7.76723706934372923710659778804, 9.29884663764465266199530893721, 10.53139943547400756053466458874, 11.58099496343850519725192023478, 13.371758569461209737297196278455, 14.24814082062726754330035283036, 15.03378842215647261412888628684, 16.26503435417022408655909980509, 16.93417650213177324134627779475, 18.20760305123965228798161100171, 19.886843731859924000022633970274, 20.99437716698912088766852827577, 22.010709145534190433433606126770, 22.75963665837833998558811909710, 23.43598637574613998894582148443, 25.30982002444757573742697930570, 25.67610721705136046553086857862, 27.01789398694630244084878662981, 27.27436021585686553243454181382, 29.412366066674303243825821930673, 30.005984228238276032927062322683

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