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Properties

Label 1-31-31.5-r0-0-0
Degree $1$
Conductor $31$
Sign $0.920 - 0.390i$
Analytic cond. $0.143963$
Root an. cond. $0.143963$
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  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + 15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + 15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(0.143963\)
Root analytic conductor: \(0.143963\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (0:\ ),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9917234476 - 0.2016130291i\)
\(L(\frac12)\) \(\approx\) \(0.9917234476 - 0.2016130291i\)
\(L(1)\) \(\approx\) \(1.240671185 - 0.1854300076i\)
\(L(1)\) \(\approx\) \(1.240671185 - 0.1854300076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
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 \( 1 + T \)
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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

−37.33763124507951456139169002909, −35.07493508255997304701971288402, −34.43472606397680543302203817494, −32.84480026430471877092839723244, −32.070402168470195074336813994, −31.27935427571917218139727539861, −29.275935871354056141906089735341, −28.47855739438873036937761989134, −27.14249440629331169185613807683, −25.367961509655525296214988188539, −24.07607760268602173405435889991, −22.89428861343269541840459684345, −21.75108813167400367032016813059, −20.79922089553807679088527996937, −19.39189449480213498659389189287, −16.96443307551322204617624730782, −15.855916310257136583308179102154, −15.06330832615849843452350368762, −12.941053661702385033080799036879, −11.940633733963307846051250648172, −10.4736777818424201699635819570, −8.49308355399650546417942111184, −6.00377840450955146839234594244, −4.911512389416397885984193439848, −3.29454214026950708006710103249, 2.590820074205882161844918063107, 4.630858894611982844454707337846, 6.78320424777052666414450693224, 7.23999033297405780573052354160, 10.569796784764362896707491962169, 11.72610737265224325593747378597, 13.06131133056759601542825206781, 14.21992737552275103556242252458, 15.77836103392387546325056024728, 17.29910378250218636585760809374, 19.026223993795233973649616148756, 20.07712135334185270980308976909, 22.00742529867724024084657866037, 23.13350962343005598496949403759, 23.627176585186930127426812210530, 25.21347400722384746118366346924, 26.514296748308603431617926427243, 28.64106656782247222335281687095, 29.614034399892768757531768916595, 30.59116155138711979558363134740, 31.4890473033686868952303133649, 33.322876688678578871016564045725, 34.08210195467783654829205674115, 35.2800601139862806956393809046, 36.583278856541551349329217440538

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