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Properties

Label 1-107-107.3-r0-0-0
Degree $1$
Conductor $107$
Sign $0.431 + 0.902i$
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  + (−0.533 − 0.845i)2-s + (0.147 + 0.989i)3-s + (−0.430 + 0.902i)4-s + (0.972 + 0.234i)5-s + (0.757 − 0.652i)6-s + (−0.794 + 0.606i)7-s + (0.992 − 0.118i)8-s + (−0.956 + 0.292i)9-s + (−0.320 − 0.947i)10-s + (−0.984 − 0.176i)11-s + (−0.956 − 0.292i)12-s + (0.0296 + 0.999i)13-s + (0.937 + 0.348i)14-s + (−0.0887 + 0.996i)15-s + (−0.630 − 0.776i)16-s + (0.582 + 0.812i)17-s + ⋯
L(s)  = 1  + (−0.533 − 0.845i)2-s + (0.147 + 0.989i)3-s + (−0.430 + 0.902i)4-s + (0.972 + 0.234i)5-s + (0.757 − 0.652i)6-s + (−0.794 + 0.606i)7-s + (0.992 − 0.118i)8-s + (−0.956 + 0.292i)9-s + (−0.320 − 0.947i)10-s + (−0.984 − 0.176i)11-s + (−0.956 − 0.292i)12-s + (0.0296 + 0.999i)13-s + (0.937 + 0.348i)14-s + (−0.0887 + 0.996i)15-s + (−0.630 − 0.776i)16-s + (0.582 + 0.812i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.431 + 0.902i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.431 + 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6370485574 + 0.4016218425i\)
\(L(\frac12)\) \(\approx\) \(0.6370485574 + 0.4016218425i\)
\(L(1)\) \(\approx\) \(0.7923731144 + 0.1688009906i\)
\(L(1)\) \(\approx\) \(0.7923731144 + 0.1688009906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.533 - 0.845i)T \)
3 \( 1 + (0.147 + 0.989i)T \)
5 \( 1 + (0.972 + 0.234i)T \)
7 \( 1 + (-0.794 + 0.606i)T \)
11 \( 1 + (-0.984 - 0.176i)T \)
13 \( 1 + (0.0296 + 0.999i)T \)
17 \( 1 + (0.582 + 0.812i)T \)
19 \( 1 + (-0.998 - 0.0592i)T \)
23 \( 1 + (0.937 - 0.348i)T \)
29 \( 1 + (0.674 + 0.737i)T \)
31 \( 1 + (0.482 - 0.875i)T \)
37 \( 1 + (0.829 + 0.558i)T \)
41 \( 1 + (-0.861 + 0.508i)T \)
43 \( 1 + (0.972 - 0.234i)T \)
47 \( 1 + (-0.861 - 0.508i)T \)
53 \( 1 + (-0.533 + 0.845i)T \)
59 \( 1 + (0.674 - 0.737i)T \)
61 \( 1 + (-0.794 - 0.606i)T \)
67 \( 1 + (0.992 + 0.118i)T \)
71 \( 1 + (0.147 - 0.989i)T \)
73 \( 1 + (0.375 - 0.926i)T \)
79 \( 1 + (-0.205 - 0.978i)T \)
83 \( 1 + (0.263 + 0.964i)T \)
89 \( 1 + (0.757 + 0.652i)T \)
97 \( 1 + (-0.320 - 0.947i)T \)
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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

−29.16894482353334422871287871783, −28.70876222893575825084316470349, −27.21209846831663053316982001104, −25.85840572898474232670645612083, −25.46921054065563009098611250377, −24.59940250933545022478699026281, −23.32661529135448526582914221962, −22.83125823061708114996021539922, −20.90811436084836018363000355923, −19.74601387777895930371413569256, −18.74073404931388774012262774690, −17.75608983531083624189438272278, −17.07469259133757434778332715388, −15.852584330916337950372902266852, −14.43076065790184729892565139526, −13.37992746082396930747332747241, −12.81266184856926959240977450780, −10.61067004830156134072469558387, −9.65149579211176037237499961559, −8.35013227526222151688619570833, −7.260502961617611007629769461971, −6.23927454735988086076674388081, −5.19711909543686760463144462016, −2.66993278530947208305253927419, −0.890373353073936187505146783357, 2.257738372475846391561349487825, 3.23784362683429279955326013127, 4.82447759030551080439848112881, 6.31505936497330013638623538735, 8.39477750030691568109141430026, 9.35027456159406090531746432663, 10.17052250892682321295893082092, 11.03601365950921015084099412104, 12.58191993524787835816442035207, 13.615047446284294342556363923706, 14.97935050216084242400800442570, 16.38991982268225222589037164911, 17.120706670932665035459823609890, 18.540393634797313693180220714606, 19.28867086052800907306596963585, 20.73991903314572952316835823605, 21.471675499052586996336495816203, 21.97284065564228268533935518380, 23.238570761311334504671592992, 25.35569565023649197126292574378, 25.94315666756777768967566708478, 26.65487934031272626415819819798, 28.011818138015119380754118985241, 28.733120839042818889650315905976, 29.39054886719085367835269145594

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