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Properties

Label 1-115-115.13-r1-0-0
Degree $1$
Conductor $115$
Sign $0.950 + 0.309i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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  + (0.989 + 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (0.540 − 0.841i)7-s + (0.909 + 0.415i)8-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.989 + 0.142i)12-s + (0.540 + 0.841i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (0.755 − 0.654i)18-s + (0.959 + 0.281i)19-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (−0.909 + 0.415i)3-s + (0.959 + 0.281i)4-s + (−0.959 + 0.281i)6-s + (0.540 − 0.841i)7-s + (0.909 + 0.415i)8-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.989 + 0.142i)12-s + (0.540 + 0.841i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (0.755 − 0.654i)18-s + (0.959 + 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.950 + 0.309i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ 0.950 + 0.309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.758855231 + 0.4381487513i\)
\(L(\frac12)\) \(\approx\) \(2.758855231 + 0.4381487513i\)
\(L(1)\) \(\approx\) \(1.720125876 + 0.2257620226i\)
\(L(1)\) \(\approx\) \(1.720125876 + 0.2257620226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 + (-0.909 + 0.415i)T \)
7 \( 1 + (0.540 - 0.841i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.989 + 0.142i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.281 - 0.959i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−28.92291593680881884026772807670, −28.36738137864098005699382041898, −27.329162849357931697741396975337, −25.27442735703071926180034974274, −24.85974727605347986150432382879, −23.626853720400878595249765340548, −22.872016795826198860143684171064, −22.04708204175794694285499781898, −20.9995645212986856534890879861, −19.92911453481640320785720680723, −18.426992501697082159336505476112, −17.69644713963558546907100710874, −16.1271189627857053615828903807, −15.39097371670412448232951762232, −14.06466009928982789975316557993, −12.84089689954703301628168139997, −12.03807753719764429105744913955, −11.20479648338005013219565859528, −9.95853498077745230892774878054, −7.8644736257097179078602209457, −6.71219314862162225170296447995, −5.43679096084353175491841876747, −4.78531450237100148597263136256, −2.80050193627995093985566475148, −1.32300238008629431554583805252, 1.26589440132915264756451847978, 3.54214721561077202187967566829, 4.504465445549450310932510005452, 5.73030040773395435773952073590, 6.70241342381941409702131057453, 8.12034898946739774411787525792, 10.16053207465530470997589487636, 11.1662446742438440856550033013, 11.8658882314747238434662324234, 13.321476443927135410182940833129, 14.21368794228368475219915287164, 15.50672740066571137587039353092, 16.52188823079766527414491657179, 17.149910185724753154088460213874, 18.679591754304810316360680287337, 20.23978420066603520419374316933, 21.24898575495171549457266058037, 21.856028550240879699725670907705, 23.13250961572717695955263080787, 23.72962859090736410705157818046, 24.52514353723882726226697661447, 26.17264749241244581118975417890, 26.93905886738825923946076078473, 28.34341563720400407443807428879, 29.212874281456568270922162194470

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