Properties

Label 1-1305-1305.38-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.940 - 0.339i$
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  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.294 − 0.955i)7-s + (0.433 − 0.900i)8-s + (0.826 + 0.563i)11-s + (0.997 + 0.0747i)13-s + (0.988 − 0.149i)14-s + (0.826 + 0.563i)16-s − i·17-s + (−0.222 − 0.974i)19-s + (−0.680 + 0.733i)22-s + (0.149 + 0.988i)23-s + (−0.222 + 0.974i)26-s + i·28-s + (−0.365 − 0.930i)31-s + (−0.680 + 0.733i)32-s + ⋯
L(s)  = 1  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (−0.294 − 0.955i)7-s + (0.433 − 0.900i)8-s + (0.826 + 0.563i)11-s + (0.997 + 0.0747i)13-s + (0.988 − 0.149i)14-s + (0.826 + 0.563i)16-s − i·17-s + (−0.222 − 0.974i)19-s + (−0.680 + 0.733i)22-s + (0.149 + 0.988i)23-s + (−0.222 + 0.974i)26-s + i·28-s + (−0.365 − 0.930i)31-s + (−0.680 + 0.733i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132972196 - 0.1979340204i\)
\(L(\frac12)\) \(\approx\) \(1.132972196 - 0.1979340204i\)
\(L(1)\) \(\approx\) \(0.9017523266 + 0.1874565581i\)
\(L(1)\) \(\approx\) \(0.9017523266 + 0.1874565581i\)

Euler product

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

−21.01673808391443795141516418852, −20.26825246297355781506059597567, −19.43268062557545111717953996889, −18.70005610936750693705493624695, −18.41684045620001501794262881610, −17.2699732301092861080451966360, −16.64084164660060309144286677451, −15.69333408556378057558170717522, −14.66476132557539740352420169636, −14.04913768230695425671215379153, −13.02873607308118432720148376853, −12.43801720359921127343505853712, −11.75315208172308444955871178922, −10.89638792709318903177753739037, −10.24660157453632608472893592148, −9.19770404445819856474716739440, −8.62811498797618839099275436821, −8.05353674757171802623981793802, −6.42594557352165756511831946853, −5.89245672321279334104704378335, −4.751786089590885941648006401579, −3.70227232154446861935897818273, −3.16123508291375620758088561942, −1.96860280598359986385788837349, −1.186733550018435378658601437163, 0.55010761072002866790965785204, 1.617511371154876667905689266307, 3.32113387581393182665152647416, 4.11142474285307513946553388422, 4.86955284502356544119202572383, 5.94467719379473124154982840502, 6.80827298193981029994267475071, 7.26281423538738352417516938809, 8.19340558562718983035879115001, 9.298982719737556491744193443720, 9.5806360013665993725990860107, 10.737753993776963099896702213062, 11.51870491383706144744342694053, 12.72474909741120608747890642711, 13.64915198527534703612738915968, 13.8150232218757015841693439496, 15.0217366222734082037465506741, 15.52005255827564523135785645940, 16.58430901637935850715796603390, 16.85182503588702123575376936252, 17.876387865741534524080394894709, 18.321274646862113991457371440721, 19.49223970206325029428042592936, 19.92739539100703010743525332830, 20.91158898225236094140871085111

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