Properties

Label 1-1305-1305.758-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.999 - 0.0380i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (0.997 − 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.149 + 0.988i)7-s + (0.974 − 0.222i)8-s + (0.955 − 0.294i)11-s + (0.680 − 0.733i)13-s + (−0.0747 + 0.997i)14-s + (0.955 − 0.294i)16-s − i·17-s + (0.623 + 0.781i)19-s + (0.930 − 0.365i)22-s + (−0.997 − 0.0747i)23-s + (0.623 − 0.781i)26-s + i·28-s + (−0.826 + 0.563i)31-s + (0.930 − 0.365i)32-s + ⋯
L(s)  = 1  + (0.997 − 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.149 + 0.988i)7-s + (0.974 − 0.222i)8-s + (0.955 − 0.294i)11-s + (0.680 − 0.733i)13-s + (−0.0747 + 0.997i)14-s + (0.955 − 0.294i)16-s − i·17-s + (0.623 + 0.781i)19-s + (0.930 − 0.365i)22-s + (−0.997 − 0.0747i)23-s + (0.623 − 0.781i)26-s + i·28-s + (−0.826 + 0.563i)31-s + (0.930 − 0.365i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.999 - 0.0380i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (758, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.999 - 0.0380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.419502619 - 0.06505256691i\)
\(L(\frac12)\) \(\approx\) \(3.419502619 - 0.06505256691i\)
\(L(1)\) \(\approx\) \(2.143672384 + 0.03091334294i\)
\(L(1)\) \(\approx\) \(2.143672384 + 0.03091334294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.997 - 0.0747i)T \)
7 \( 1 + (-0.149 + 0.988i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (0.680 - 0.733i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (-0.997 - 0.0747i)T \)
31 \( 1 + (-0.826 + 0.563i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.563 - 0.826i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (-0.433 + 0.900i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.294 - 0.955i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.433 + 0.900i)T \)
79 \( 1 + (-0.733 + 0.680i)T \)
83 \( 1 + (-0.930 - 0.365i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.930 + 0.365i)T \)
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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

−21.016464261210097254670987333800, −20.22749088426861022232619739480, −19.777515154915646029626218272413, −18.958448114923652731319944488228, −17.66871645016546730344332759188, −17.024200117536730961771095912272, −16.267988079109561156502508664480, −15.646403347223239655063904433803, −14.48180832121196249111281517513, −14.21563943410080378556984179184, −13.26052429357797230291490991065, −12.72473115396110003908520351100, −11.5772550495771395920000031189, −11.21008364167261628912916413423, −10.1968332318328509567814715587, −9.336532786676846505373409013195, −8.13809786037410069038891451866, −7.27766288970036938050915852854, −6.5325055734405141405114891747, −5.922138802287945858370339061299, −4.64986399314915295278701954709, −4.00056849268260035324560572955, −3.427764648361170805442259003163, −2.04927871605705091227998309695, −1.21244099430447623444467721452, 1.17535019789060185849103701655, 2.2249421312848584118915143220, 3.20317760716542275870054940988, 3.83268591854411222039974937905, 5.00396082420050712481268103985, 5.78349286972686568177033386982, 6.302649714148585052868722163193, 7.3674453913686298033980200859, 8.28948150660906688024374292758, 9.268934154117961338459559006584, 10.14154510043378639521532146238, 11.18412890121324098043238962766, 11.84227504503569576609694338143, 12.39216576269500488043735734606, 13.29314064926808617403926274388, 14.10420276815298784539419796905, 14.68069883526783681854362776706, 15.62629548344740165667710907938, 16.1086980943074663205463073065, 16.88704358077629689286235046443, 18.19992660059489671623415872700, 18.59372281480309900973509930338, 19.83767605572536709871486205622, 20.16771342645541698958889466975, 21.13903384121136198294018889417

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