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Properties

Label 1-55e2-3025.522-r1-0-0
Degree $1$
Conductor $3025$
Sign $0.106 + 0.994i$
Analytic cond. $325.081$
Root an. cond. $325.081$
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  + (−0.755 − 0.654i)2-s + (0.587 + 0.809i)3-s + (0.142 + 0.989i)4-s + (0.0855 − 0.996i)6-s + (−0.825 − 0.564i)7-s + (0.540 − 0.841i)8-s + (−0.309 + 0.951i)9-s + (−0.717 + 0.696i)12-s + (−0.170 + 0.985i)13-s + (0.254 + 0.967i)14-s + (−0.959 + 0.281i)16-s + (−0.980 + 0.198i)17-s + (0.856 − 0.516i)18-s + (−0.415 − 0.909i)19-s + (−0.0285 − 0.999i)21-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (0.587 + 0.809i)3-s + (0.142 + 0.989i)4-s + (0.0855 − 0.996i)6-s + (−0.825 − 0.564i)7-s + (0.540 − 0.841i)8-s + (−0.309 + 0.951i)9-s + (−0.717 + 0.696i)12-s + (−0.170 + 0.985i)13-s + (0.254 + 0.967i)14-s + (−0.959 + 0.281i)16-s + (−0.980 + 0.198i)17-s + (0.856 − 0.516i)18-s + (−0.415 − 0.909i)19-s + (−0.0285 − 0.999i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.106 + 0.994i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ 0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4906842862 + 0.4408154911i\)
\(L(\frac12)\) \(\approx\) \(0.4906842862 + 0.4408154911i\)
\(L(1)\) \(\approx\) \(0.6877140329 + 0.009004629804i\)
\(L(1)\) \(\approx\) \(0.6877140329 + 0.009004629804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (-0.825 - 0.564i)T \)
13 \( 1 + (-0.170 + 0.985i)T \)
17 \( 1 + (-0.980 + 0.198i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.336 - 0.941i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.985 + 0.170i)T \)
37 \( 1 + (-0.170 - 0.985i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (-0.226 + 0.974i)T \)
53 \( 1 + (0.791 - 0.610i)T \)
59 \( 1 + (-0.516 + 0.856i)T \)
61 \( 1 + (0.516 - 0.856i)T \)
67 \( 1 + (-0.856 + 0.516i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.791 - 0.610i)T \)
79 \( 1 + (0.998 + 0.0570i)T \)
83 \( 1 + (-0.336 - 0.941i)T \)
89 \( 1 + (0.870 - 0.491i)T \)
97 \( 1 + (0.931 - 0.362i)T \)
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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

−18.58474900054787266513020126562, −18.13526081274252219133745054026, −17.48206309152684882471759108201, −16.66013985203698056217477147400, −15.9084381466276970974489778160, −15.11473922614073440081012590294, −14.82953361258806652831367020362, −13.82508715790080937734533194815, −13.106203918609745905147314123170, −12.580939350729881388788275944700, −11.62356934285874921483930933028, −10.75986686997300634410356187452, −9.86795815216953370414182982463, −9.2060897699859283563594233667, −8.68075019447391841310417432899, −7.87545850131875495882832971422, −7.267310733643508742937422879518, −6.51750388669431141128288190924, −5.883924141141465452439464030289, −5.18279684193254428430902951889, −3.78287647506280412966009778480, −2.8761947616592829912195474690, −2.10563191237390159203563490814, −1.22051165060064628073572457735, −0.19630402899098638899560233287, 0.57657996452608637214164689448, 2.06810014055384579858712581540, 2.47522749788658069957826707472, 3.50264006006933758117498260046, 4.14745801613068343736803834929, 4.66957760030381023567720763417, 6.13940741070899681603905256184, 7.02260563253988182430561335887, 7.58750894917420956257843658959, 8.741131991652658805047419186609, 9.09915649628595882406158177639, 9.62557853076450715178060991932, 10.67396766033104829308832793780, 10.8038622402141014141960661704, 11.74984105506367388553950636632, 12.81647600639170032564618623919, 13.26244888513175946529389531316, 14.06119357358967293172552860860, 14.90769496859089143305352669949, 15.89050287631943141707630492915, 16.194872559672786144735939005987, 17.018871671637884373399191767789, 17.48589617508514920913437936680, 18.60465519274642859469348420317, 19.268396698146321949841834140911

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