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Properties

Label 1-1007-1007.17-r0-0-0
Degree $1$
Conductor $1007$
Sign $-0.757 - 0.653i$
Analytic cond. $4.67648$
Root an. cond. $4.67648$
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  + (−0.0134 − 0.999i)2-s + (−0.998 + 0.0536i)3-s + (−0.999 + 0.0268i)4-s + (0.897 − 0.440i)5-s + (0.0670 + 0.997i)6-s + (0.987 + 0.160i)7-s + (0.0402 + 0.999i)8-s + (0.994 − 0.107i)9-s + (−0.452 − 0.891i)10-s + (0.428 − 0.903i)11-s + (0.996 − 0.0804i)12-s + (−0.783 + 0.621i)13-s + (0.147 − 0.989i)14-s + (−0.872 + 0.488i)15-s + (0.998 − 0.0536i)16-s + (−0.991 + 0.133i)17-s + ⋯
L(s)  = 1  + (−0.0134 − 0.999i)2-s + (−0.998 + 0.0536i)3-s + (−0.999 + 0.0268i)4-s + (0.897 − 0.440i)5-s + (0.0670 + 0.997i)6-s + (0.987 + 0.160i)7-s + (0.0402 + 0.999i)8-s + (0.994 − 0.107i)9-s + (−0.452 − 0.891i)10-s + (0.428 − 0.903i)11-s + (0.996 − 0.0804i)12-s + (−0.783 + 0.621i)13-s + (0.147 − 0.989i)14-s + (−0.872 + 0.488i)15-s + (0.998 − 0.0536i)16-s + (−0.991 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $-0.757 - 0.653i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ -0.757 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3740077259 - 1.006356319i\)
\(L(\frac12)\) \(\approx\) \(0.3740077259 - 1.006356319i\)
\(L(1)\) \(\approx\) \(0.7005434677 - 0.5289247559i\)
\(L(1)\) \(\approx\) \(0.7005434677 - 0.5289247559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.0134 - 0.999i)T \)
3 \( 1 + (-0.998 + 0.0536i)T \)
5 \( 1 + (0.897 - 0.440i)T \)
7 \( 1 + (0.987 + 0.160i)T \)
11 \( 1 + (0.428 - 0.903i)T \)
13 \( 1 + (-0.783 + 0.621i)T \)
17 \( 1 + (-0.991 + 0.133i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.252 + 0.967i)T \)
31 \( 1 + (-0.428 - 0.903i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (0.711 - 0.702i)T \)
43 \( 1 + (0.730 + 0.682i)T \)
47 \( 1 + (0.830 - 0.556i)T \)
59 \( 1 + (0.994 + 0.107i)T \)
61 \( 1 + (-0.379 - 0.925i)T \)
67 \( 1 + (-0.476 + 0.879i)T \)
71 \( 1 + (-0.730 - 0.682i)T \)
73 \( 1 + (-0.611 - 0.791i)T \)
79 \( 1 + (0.982 + 0.186i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.303 - 0.952i)T \)
97 \( 1 + (0.830 + 0.556i)T \)
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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

−22.2104290061590878331248422060, −21.55868347965321825760106468416, −20.54916355972219292205803575330, −19.36040654110582853777383308418, −18.21757466609501394316809115204, −17.7053429227805349906937866372, −17.4606119580311672957733172257, −16.71179152056641076834827049102, −15.54296499840034490443790759250, −14.98751089393529199235429832790, −14.14789379097467438862103389929, −13.36068704461489130668171467929, −12.506513141851705231211501919161, −11.5793607791806645928231319774, −10.52052157205849673283491515588, −9.90674776795674631482025985067, −9.0739420419012492674685935776, −7.73143918454401712264760732494, −7.17765174283218934090701100181, −6.336568151626710826566464479724, −5.53387206928223135353657324338, −4.83133417353491576673403605721, −4.09115515703150893027727851078, −2.25887678579567423726224589124, −1.19330148504051165533122421148, 0.58981746790541832635613727075, 1.71974739254044573008564530234, 2.31901184235711645455576406634, 4.03353840945610203857259044865, 4.65103648157633271260137840506, 5.50055286540556771553517760005, 6.18388489978392657400154641516, 7.5190739911513251922271213499, 8.814540667045690109867502950847, 9.24206691950568634455702228342, 10.35486496026684476561028779615, 10.96680735650576157645167820765, 11.674602680780341845308417666998, 12.40001884911918931887397553279, 13.17241200853319608967187978927, 14.08290652104473881626123681530, 14.6870214693345063404491563758, 16.19078796623955607814950945141, 16.91450329681359725838670268821, 17.57714148713746829260294846915, 18.10308320542778707353958764126, 18.88410820937939761262810867472, 19.88470525016967263709855449434, 20.7972740099278864566099858395, 21.40433007787686995142813554903

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