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Properties

Label 1-389-389.36-r0-0-0
Degree $1$
Conductor $389$
Sign $0.906 - 0.423i$
Analytic cond. $1.80650$
Root an. cond. $1.80650$
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  + (−0.816 + 0.577i)2-s + (0.966 − 0.256i)3-s + (0.333 − 0.942i)4-s + (0.145 − 0.989i)5-s + (−0.641 + 0.767i)6-s + (0.712 − 0.701i)7-s + (0.271 + 0.962i)8-s + (0.868 − 0.495i)9-s + (0.452 + 0.891i)10-s + (0.563 + 0.825i)11-s + (0.0808 − 0.996i)12-s + (0.665 − 0.746i)13-s + (−0.177 + 0.984i)14-s + (−0.113 − 0.993i)15-s + (−0.777 − 0.628i)16-s + (0.452 + 0.891i)17-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)2-s + (0.966 − 0.256i)3-s + (0.333 − 0.942i)4-s + (0.145 − 0.989i)5-s + (−0.641 + 0.767i)6-s + (0.712 − 0.701i)7-s + (0.271 + 0.962i)8-s + (0.868 − 0.495i)9-s + (0.452 + 0.891i)10-s + (0.563 + 0.825i)11-s + (0.0808 − 0.996i)12-s + (0.665 − 0.746i)13-s + (−0.177 + 0.984i)14-s + (−0.113 − 0.993i)15-s + (−0.777 − 0.628i)16-s + (0.452 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(389\)
Sign: $0.906 - 0.423i$
Analytic conductor: \(1.80650\)
Root analytic conductor: \(1.80650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{389} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 389,\ (0:\ ),\ 0.906 - 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449762817 - 0.3218648473i\)
\(L(\frac12)\) \(\approx\) \(1.449762817 - 0.3218648473i\)
\(L(1)\) \(\approx\) \(1.172787414 - 0.09558642535i\)
\(L(1)\) \(\approx\) \(1.172787414 - 0.09558642535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad389 \( 1 \)
good2 \( 1 + (-0.816 + 0.577i)T \)
3 \( 1 + (0.966 - 0.256i)T \)
5 \( 1 + (0.145 - 0.989i)T \)
7 \( 1 + (0.712 - 0.701i)T \)
11 \( 1 + (0.563 + 0.825i)T \)
13 \( 1 + (0.665 - 0.746i)T \)
17 \( 1 + (0.452 + 0.891i)T \)
19 \( 1 + (-0.536 + 0.843i)T \)
23 \( 1 + (0.145 + 0.989i)T \)
29 \( 1 + (0.797 - 0.603i)T \)
31 \( 1 + (-0.852 + 0.523i)T \)
37 \( 1 + (0.509 + 0.860i)T \)
41 \( 1 + (-0.536 - 0.843i)T \)
43 \( 1 + (-0.995 + 0.0970i)T \)
47 \( 1 + (-0.536 - 0.843i)T \)
53 \( 1 + (-0.937 + 0.348i)T \)
59 \( 1 + (-0.240 + 0.970i)T \)
61 \( 1 + (0.665 - 0.746i)T \)
67 \( 1 + (-0.999 - 0.0323i)T \)
71 \( 1 + (0.333 - 0.942i)T \)
73 \( 1 + (-0.777 + 0.628i)T \)
79 \( 1 + (-0.852 - 0.523i)T \)
83 \( 1 + (-0.999 + 0.0323i)T \)
89 \( 1 + (-0.363 + 0.931i)T \)
97 \( 1 + (0.333 + 0.942i)T \)
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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

−25.02658993419263058750543999642, −23.88354231155826547702557889369, −22.28884174416910356252797782328, −21.60278954422744230806534830168, −21.112143800681409241250870397727, −20.08790758250784624331950023948, −19.111213949893205831048557482103, −18.60052582439407872969799664893, −17.95307626663871350156328971642, −16.58573957875610529825233856236, −15.76726662938328256931345811330, −14.64676812749327906745131437691, −14.04132285357426285062421256662, −12.91030690966715372045274837696, −11.48507794064183705353488095637, −11.081394012832503882246371165504, −9.926501854504213649600759157336, −8.97213973573092649322828719462, −8.44094791422218538581716269506, −7.31398568863153610587004262580, −6.35861421170490393002596314539, −4.50790762507684002910678364458, −3.30909762609207608709941830921, −2.581215972453961714749749450199, −1.55806232545851208677760583496, 1.31282293968187402369136172457, 1.7256137889973552858531172094, 3.71616118029997146601886121936, 4.806155324358245208054751130624, 6.068105983440161001541788490492, 7.27981177383445295002461812387, 8.14134514722281660811389762495, 8.591899494299130934479271682934, 9.74669689167874338352501613669, 10.40252791652219859147784831804, 11.87452927708968185670040026095, 13.00043925538512309118922224628, 13.90044257671695738478189818053, 14.804199902745780030863743000453, 15.48407534364259698256018176203, 16.66990785489737097377308712545, 17.3884809046113597571031806902, 18.118341097669990764378678740228, 19.28105739693641953389271233025, 20.05637845370049594493699970121, 20.5344095441444823014642968564, 21.37057259275626272363354072159, 23.3347650827302953715466603671, 23.65797845305165381454807454348, 24.73836949026747854627690971122

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