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Properties

Label 1-107-107.49-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.535 - 0.844i$
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  + (0.375 − 0.926i)2-s + (0.263 − 0.964i)3-s + (−0.717 − 0.696i)4-s + (0.674 + 0.737i)5-s + (−0.794 − 0.606i)6-s + (0.757 − 0.652i)7-s + (−0.915 + 0.403i)8-s + (−0.861 − 0.508i)9-s + (0.937 − 0.348i)10-s + (0.582 − 0.812i)11-s + (−0.861 + 0.508i)12-s + (−0.630 + 0.776i)13-s + (−0.320 − 0.947i)14-s + (0.889 − 0.456i)15-s + (0.0296 + 0.999i)16-s + (−0.984 − 0.176i)17-s + ⋯
L(s)  = 1  + (0.375 − 0.926i)2-s + (0.263 − 0.964i)3-s + (−0.717 − 0.696i)4-s + (0.674 + 0.737i)5-s + (−0.794 − 0.606i)6-s + (0.757 − 0.652i)7-s + (−0.915 + 0.403i)8-s + (−0.861 − 0.508i)9-s + (0.937 − 0.348i)10-s + (0.582 − 0.812i)11-s + (−0.861 + 0.508i)12-s + (−0.630 + 0.776i)13-s + (−0.320 − 0.947i)14-s + (0.889 − 0.456i)15-s + (0.0296 + 0.999i)16-s + (−0.984 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6563358896 - 1.193009024i\)
\(L(\frac12)\) \(\approx\) \(0.6563358896 - 1.193009024i\)
\(L(1)\) \(\approx\) \(0.9727085585 - 0.9267016762i\)
\(L(1)\) \(\approx\) \(0.9727085585 - 0.9267016762i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.375 - 0.926i)T \)
3 \( 1 + (0.263 - 0.964i)T \)
5 \( 1 + (0.674 + 0.737i)T \)
7 \( 1 + (0.757 - 0.652i)T \)
11 \( 1 + (0.582 - 0.812i)T \)
13 \( 1 + (-0.630 + 0.776i)T \)
17 \( 1 + (-0.984 - 0.176i)T \)
19 \( 1 + (-0.205 + 0.978i)T \)
23 \( 1 + (-0.320 + 0.947i)T \)
29 \( 1 + (0.972 - 0.234i)T \)
31 \( 1 + (0.829 - 0.558i)T \)
37 \( 1 + (0.482 - 0.875i)T \)
41 \( 1 + (-0.956 + 0.292i)T \)
43 \( 1 + (0.674 - 0.737i)T \)
47 \( 1 + (-0.956 - 0.292i)T \)
53 \( 1 + (0.375 + 0.926i)T \)
59 \( 1 + (0.972 + 0.234i)T \)
61 \( 1 + (0.757 + 0.652i)T \)
67 \( 1 + (-0.915 - 0.403i)T \)
71 \( 1 + (0.263 + 0.964i)T \)
73 \( 1 + (-0.533 + 0.845i)T \)
79 \( 1 + (-0.998 - 0.0592i)T \)
83 \( 1 + (0.147 + 0.989i)T \)
89 \( 1 + (-0.794 + 0.606i)T \)
97 \( 1 + (0.937 - 0.348i)T \)
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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

−30.41979523403970714278642378143, −28.55315193328193278652010595274, −27.7460063668020857968480797520, −26.85116614986964493852661715929, −25.618537476543707577619415779577, −24.94433696872488169200303695856, −24.10561760973251257296921023417, −22.48768634586803073611120554741, −21.861470472671402709237706879411, −20.91813101729728784108710978256, −19.887718275640832343459746494530, −17.720932947297362417426375383244, −17.381132740653541752140165991483, −16.05136612757004757630606926629, −15.1112154067392716438815176728, −14.364971333762746890608453609453, −13.08855092135311994214746750776, −11.89008971835005122157279904661, −10.06756206453029586165200211369, −8.95356599407519563396814035745, −8.25519022258664376366965367921, −6.40640924782305445248493714155, −4.948231643505960215407851762642, −4.600601304121856075003323851836, −2.55046880553451367618880742509, 1.4529246418237740157049792027, 2.513722553705606233822275554375, 4.01058962830118468084042192723, 5.78742919254558640498815547514, 6.93181228529717977821808552456, 8.49799054867866634311159067778, 9.8261394244119057247172621101, 11.150787661331035383127968726513, 11.891577562754110161865567029766, 13.46576957543302643236128409366, 13.98690532080716254087343096411, 14.75087814854813528021867310882, 17.16853338627899329012187098458, 17.95920514126548948363771600441, 18.99719243920031260403109117914, 19.76376417233465120694411202217, 20.99880570809472381554123140874, 21.91113095546955708265395061449, 23.04640759876466367421972270830, 24.04262121655173093329393772616, 24.89367053381583744713201278151, 26.44925415919002492410621340761, 27.153019083644294019142689060647, 28.80803303015742814515502850909, 29.54015200463381614110818131849

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