Properties

Label 1-317-317.316-r0-0-0
Degree $1$
Conductor $317$
Sign $1$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $1$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{317} (316, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4980976757\)
\(L(\frac12)\) \(\approx\) \(0.4980976757\)
\(L(1)\) \(\approx\) \(0.5042280493\)
\(L(1)\) \(\approx\) \(0.5042280493\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
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

−24.91703373782302531512673835847, −24.3470681196858766754339168669, −23.64622194040700147211887526710, −22.52257097032627016512829972480, −21.58112541187085586291765107745, −20.553122883842614693465982998687, −19.530120926075976143624391005428, −18.89235111531679984148542217064, −17.76742987844998005683343098564, −17.140015310929424985592355749364, −16.483295555114703463798567265915, −15.22456355147256461632596185241, −14.81733562584882632617264836261, −12.7923509949725842870613878364, −11.75191092571873434763333798184, −11.367424855853743297098072149209, −10.528208797697640719779083216939, −9.251089459101163792808448229982, −8.22865357837669369583326241481, −7.2139420551603165984640707938, −6.52442054767685302490511287128, −5.01574651419739139108916292540, −4.034367229749505618172075145461, −2.15563906127462621370826538836, −0.80522733078214625227075808520, 0.80522733078214625227075808520, 2.15563906127462621370826538836, 4.034367229749505618172075145461, 5.01574651419739139108916292540, 6.52442054767685302490511287128, 7.2139420551603165984640707938, 8.22865357837669369583326241481, 9.251089459101163792808448229982, 10.528208797697640719779083216939, 11.367424855853743297098072149209, 11.75191092571873434763333798184, 12.7923509949725842870613878364, 14.81733562584882632617264836261, 15.22456355147256461632596185241, 16.483295555114703463798567265915, 17.140015310929424985592355749364, 17.76742987844998005683343098564, 18.89235111531679984148542217064, 19.530120926075976143624391005428, 20.553122883842614693465982998687, 21.58112541187085586291765107745, 22.52257097032627016512829972480, 23.64622194040700147211887526710, 24.3470681196858766754339168669, 24.91703373782302531512673835847

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