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Properties

Label 1-107-107.64-r0-0-0
Degree $1$
Conductor $107$
Sign $0.976 + 0.217i$
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  + (0.937 + 0.348i)2-s + (0.972 − 0.234i)3-s + (0.757 + 0.652i)4-s + (−0.533 − 0.845i)5-s + (0.992 + 0.118i)6-s + (−0.915 + 0.403i)7-s + (0.482 + 0.875i)8-s + (0.889 − 0.456i)9-s + (−0.205 − 0.978i)10-s + (0.0296 + 0.999i)11-s + (0.889 + 0.456i)12-s + (0.263 − 0.964i)13-s + (−0.998 + 0.0592i)14-s + (−0.717 − 0.696i)15-s + (0.147 + 0.989i)16-s + (−0.630 − 0.776i)17-s + ⋯
L(s)  = 1  + (0.937 + 0.348i)2-s + (0.972 − 0.234i)3-s + (0.757 + 0.652i)4-s + (−0.533 − 0.845i)5-s + (0.992 + 0.118i)6-s + (−0.915 + 0.403i)7-s + (0.482 + 0.875i)8-s + (0.889 − 0.456i)9-s + (−0.205 − 0.978i)10-s + (0.0296 + 0.999i)11-s + (0.889 + 0.456i)12-s + (0.263 − 0.964i)13-s + (−0.998 + 0.0592i)14-s + (−0.717 − 0.696i)15-s + (0.147 + 0.989i)16-s + (−0.630 − 0.776i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.976 + 0.217i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.976 + 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.966321047 + 0.2163384820i\)
\(L(\frac12)\) \(\approx\) \(1.966321047 + 0.2163384820i\)
\(L(1)\) \(\approx\) \(1.883797313 + 0.1742214989i\)
\(L(1)\) \(\approx\) \(1.883797313 + 0.1742214989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.937 + 0.348i)T \)
3 \( 1 + (0.972 - 0.234i)T \)
5 \( 1 + (-0.533 - 0.845i)T \)
7 \( 1 + (-0.915 + 0.403i)T \)
11 \( 1 + (0.0296 + 0.999i)T \)
13 \( 1 + (0.263 - 0.964i)T \)
17 \( 1 + (-0.630 - 0.776i)T \)
19 \( 1 + (-0.861 + 0.508i)T \)
23 \( 1 + (-0.998 - 0.0592i)T \)
29 \( 1 + (0.375 - 0.926i)T \)
31 \( 1 + (-0.984 - 0.176i)T \)
37 \( 1 + (0.582 + 0.812i)T \)
41 \( 1 + (-0.0887 + 0.996i)T \)
43 \( 1 + (-0.533 + 0.845i)T \)
47 \( 1 + (-0.0887 - 0.996i)T \)
53 \( 1 + (0.937 - 0.348i)T \)
59 \( 1 + (0.375 + 0.926i)T \)
61 \( 1 + (-0.915 - 0.403i)T \)
67 \( 1 + (0.482 - 0.875i)T \)
71 \( 1 + (0.972 + 0.234i)T \)
73 \( 1 + (-0.320 - 0.947i)T \)
79 \( 1 + (-0.956 - 0.292i)T \)
83 \( 1 + (0.674 + 0.737i)T \)
89 \( 1 + (0.992 - 0.118i)T \)
97 \( 1 + (-0.205 - 0.978i)T \)
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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

−29.934251922473324232816426963318, −28.91246642712170486919303329707, −27.44346610696450948149195133985, −26.23722723590157104034849332382, −25.70939497153090362675170111372, −24.14160398495317337932975740873, −23.46333343952846020507806162940, −21.98464394034971660620439218687, −21.646835708458146747361815504570, −20.0793183174410967547534869885, −19.42603978600383886510826830678, −18.72354562447298018340820159596, −16.33621508839808395220703942830, −15.6038620979605777168021222953, −14.45624269911970066434029198860, −13.73810138649045227503620047235, −12.67276760327297111121112726804, −11.11205057523317736284172218335, −10.32315801265627238332464493999, −8.83749999065142610060791588071, −7.18780448117942437456334480876, −6.26359319068673703643423116157, −4.10801939285798824903729889809, −3.51211342304621046296299801394, −2.24624264250823708658108580670, 2.21383330816757997101283217985, 3.55457669408595061456320015730, 4.64648221883272776466783648706, 6.26629830493413119494232112772, 7.57133018115449413617227104026, 8.52326945744299478199339126622, 9.881093320686282979657197253565, 11.92300770771334511435734061451, 12.80261597759256812828079122762, 13.39409692741821953275521108561, 14.973681577287829942624572936679, 15.54779932889911369365141158890, 16.56527360841155760837026740556, 18.179962717416313036888190730832, 19.84841510309408884312293084332, 20.145701535574333819727424075971, 21.292837376970040952104888743797, 22.63656921519468831620790554139, 23.48059116825951309569202016027, 24.740351961425576685890424670559, 25.23436075561433662187915687447, 26.1506561756952399707348619795, 27.569028050424255929369967445819, 28.86248049695512889732174844002, 29.99567004825215290017647782325

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