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Properties

Label 1-315-315.194-r0-0-0
Degree $1$
Conductor $315$
Sign $0.998 + 0.0477i$
Analytic cond. $1.46285$
Root an. cond. $1.46285$
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 + 4-s + 8-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)26-s + (0.5 − 0.866i)29-s − 31-s + 32-s + (0.5 − 0.866i)34-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)26-s + (0.5 − 0.866i)29-s − 31-s + 32-s + (0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.998 + 0.0477i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ 0.998 + 0.0477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.448763500 + 0.05851518286i\)
\(L(\frac12)\) \(\approx\) \(2.448763500 + 0.05851518286i\)
\(L(1)\) \(\approx\) \(1.942853392 + 0.02430229528i\)
\(L(1)\) \(\approx\) \(1.942853392 + 0.02430229528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + T \)
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 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
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 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−24.92175787660203178741855834679, −24.086420447293973037318804914273, −23.59880242257406479439765600658, −22.29539455682824831689004169875, −21.799550427351492635729818421175, −20.99984777139991453202085184608, −19.79089341868661359675136648894, −19.29902829873082287317737458940, −17.94537969960638072081698314881, −16.55648893216962729121964266936, −16.27633436876343998462965586741, −14.80441778480227319685197637169, −14.33139367835263622215467194468, −13.30879209330825701037299535405, −12.359029746964301940868490216266, −11.48178190125780921493780038383, −10.641079500600462859321704527072, −9.336918538674050282878851961862, −8.08201766141650057688345764479, −6.888382772195045499515109404465, −6.07827429140428782057491793372, −4.937045909953657391728026727957, −3.910197225928375079231412067408, −2.8528637916270610692603108592, −1.514372705216294414613439082, 1.562971881272407669200145838029, 2.84704131605094816054052148209, 3.915160606163326765738630866140, 5.04599220316606267866963662490, 5.90031663061981789714147372592, 7.17647545984247038669930108672, 7.854838544562919907632584369353, 9.59055928910619625963963563551, 10.38322653635253166321336555546, 11.76270126011382920788071684346, 12.22668192423101375446094084042, 13.33075476096421130083764733174, 14.246463118883203218777710576799, 15.048284952460447757827192649690, 15.903280338067767897124890942464, 16.911082150494808160656301352204, 17.88050091767694252740507729175, 19.15712109338846584475676864372, 20.2040982750526306488917928424, 20.6405857337640609785667396580, 21.86246280496920654936059198094, 22.58094071013966089464339421929, 23.22766556528485629106510853125, 24.262195241817523748459913118923, 25.19704151787961808015819010489

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