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Properties

Label 1-1037-1037.135-r0-0-0
Degree $1$
Conductor $1037$
Sign $-0.876 - 0.481i$
Analytic cond. $4.81580$
Root an. cond. $4.81580$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + 8-s + 9-s + (0.5 − 0.866i)10-s − 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + 8-s + 9-s + (0.5 − 0.866i)10-s − 11-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04442669874 - 0.1731779916i\)
\(L(\frac12)\) \(\approx\) \(0.04442669874 - 0.1731779916i\)
\(L(1)\) \(\approx\) \(0.5084872731 - 0.08200962354i\)
\(L(1)\) \(\approx\) \(0.5084872731 - 0.08200962354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−21.79422454787202208449258690258, −21.34118969173223778859606623010, −20.235066976182649381531230200472, −19.46314821851617234643089855369, −18.33548565205432865238674700793, −17.7444210796982202321462517552, −17.21637417632002100733400247338, −16.44922560003721577941057607563, −16.05313158526891394685902145149, −15.03296661176523632884045840979, −13.88450872703285669190750763872, −13.40717725909848428103871076967, −12.42382830549167303734177221029, −11.41725724812474722539449381226, −10.31219271636090390787101248320, −10.10106530573559536195419518888, −8.8856470253770864723612737051, −8.09911978697381618365229990168, −7.07715382028987306156826621793, −6.52860195119963201212198550128, −5.32230882249936370484588370460, −4.91558598354860353567595933077, −4.19366872101961484674823020375, −1.977337166925887624117625305238, −1.09910454368745912158801196650, 0.112052407869239873093908896950, 1.80020095287434719431094350463, 2.39256418623303481114380660016, 3.490505182979901624727173701697, 4.766443581054447539174721012075, 5.54426130764136077001248695690, 6.38573958157241764538999367657, 7.6622108376857395687150499241, 8.171436426294458256473261150641, 9.56509485767751778807798284220, 10.31628611825163799010336105662, 10.61916854021238022883962516956, 11.694776798173201921457835182751, 12.18421563871229559608424539835, 13.04990095094124113500777057435, 13.90816937668913767413283821242, 15.12741212125452043403341249977, 15.75173648255300966251840525402, 16.99026978536699607319014696599, 17.52225860420175229263950380323, 18.35114213850357224937970146101, 18.511186057861488647853350684133, 19.44249191217117902980012791052, 20.81285429743825290211587590960, 21.17500687817776882807706552283

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