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Properties

Label 1-1513-1513.764-r1-0-0
Degree $1$
Conductor $1513$
Sign $-0.632 + 0.774i$
Analytic cond. $162.594$
Root an. cond. $162.594$
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 + (0.707 + 0.707i)3-s + 4-s + i·5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + 8-s + i·9-s + i·10-s + 11-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s + ⋯
L(s)  = 1  + 2-s + (0.707 + 0.707i)3-s + 4-s + i·5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + 8-s + i·9-s + i·10-s + 11-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1513 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1513 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1513\)    =    \(17 \cdot 89\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(162.594\)
Root analytic conductor: \(162.594\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1513} (764, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1513,\ (1:\ ),\ -0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.271529880 + 6.893632778i\)
\(L(\frac12)\) \(\approx\) \(3.271529880 + 6.893632778i\)
\(L(1)\) \(\approx\) \(2.463720311 + 1.690932774i\)
\(L(1)\) \(\approx\) \(2.463720311 + 1.690932774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
89 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + iT \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + (0.707 + 0.707i)T \)
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

−20.27682317110328638862359426937, −19.73111562912599097239942632021, −18.99978826079563624324546378337, −17.67257103740530718409661499912, −17.12382374933458132210661626746, −16.41346786940553189923127994763, −15.30635734077723926314010134483, −14.68702784047900653636299798249, −13.93375307965116819504898795170, −13.51991313282501013067481958281, −12.41788326543840224373755135766, −12.26063265079384965588845840475, −11.30630187835008586261730595333, −10.21261992275578769749098641435, −9.2539823024509457463857458704, −8.2627602999083840011660790998, −7.632238578535878938574724028927, −6.94254226567805536663948742930, −5.92188739263129641901048088024, −5.02554547799116987752565261038, −4.20334206592993255127604068971, −3.48713868804836326066684945116, −2.3763803974220368486866441685, −1.39772092808902375829466404402, −0.85042767610295850756114197300, 1.66077769749226288042046253398, 2.3547419989887621514325282221, 3.19818153487098003432246569876, 3.88797494816627901451881199363, 4.85860305222686300981474498563, 5.4609169687889193578377789584, 6.692403748725531900552080850789, 7.25085150252592807587590794115, 8.21734040827253785335009163018, 9.28729477137342102341854463149, 9.91984845564103047391535870476, 11.12946892314541376268343084657, 11.39290097572355354703471151416, 12.22124967004091332022818627072, 13.49884049491564712172387984180, 14.07275255803828671117262922897, 14.67650864415068905720110024750, 15.15309954117628660277225587206, 15.74233441795512739503336757959, 16.799436413286206890771093390224, 17.52532075445088584516954262097, 18.83120872369509681430607714306, 19.29572081234181637509075129512, 20.08764121593841800410756766613, 20.90736917907620906725563147503

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