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Properties

Label 1-4025-4025.6-r1-0-0
Degree $1$
Conductor $4025$
Sign $-0.960 + 0.279i$
Analytic cond. $432.546$
Root an. cond. $432.546$
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  + (0.974 + 0.226i)2-s + (−0.774 + 0.633i)3-s + (0.897 + 0.441i)4-s + (−0.897 + 0.441i)6-s + (0.774 + 0.633i)8-s + (0.198 − 0.980i)9-s + (0.0855 − 0.996i)11-s + (−0.974 + 0.226i)12-s + (−0.941 − 0.336i)13-s + (0.610 + 0.791i)16-s + (−0.897 + 0.441i)17-s + (0.415 − 0.909i)18-s + (0.985 − 0.170i)19-s + (0.309 − 0.951i)22-s − 24-s + ⋯
L(s)  = 1  + (0.974 + 0.226i)2-s + (−0.774 + 0.633i)3-s + (0.897 + 0.441i)4-s + (−0.897 + 0.441i)6-s + (0.774 + 0.633i)8-s + (0.198 − 0.980i)9-s + (0.0855 − 0.996i)11-s + (−0.974 + 0.226i)12-s + (−0.941 − 0.336i)13-s + (0.610 + 0.791i)16-s + (−0.897 + 0.441i)17-s + (0.415 − 0.909i)18-s + (0.985 − 0.170i)19-s + (0.309 − 0.951i)22-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.960 + 0.279i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ -0.960 + 0.279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1815269149 + 1.272534477i\)
\(L(\frac12)\) \(\approx\) \(0.1815269149 + 1.272534477i\)
\(L(1)\) \(\approx\) \(1.263646367 + 0.3991999919i\)
\(L(1)\) \(\approx\) \(1.263646367 + 0.3991999919i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.974 + 0.226i)T \)
3 \( 1 + (-0.774 + 0.633i)T \)
11 \( 1 + (0.0855 - 0.996i)T \)
13 \( 1 + (-0.941 - 0.336i)T \)
17 \( 1 + (-0.897 + 0.441i)T \)
19 \( 1 + (0.985 - 0.170i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (-0.774 - 0.633i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.564 + 0.825i)T \)
59 \( 1 + (-0.941 - 0.336i)T \)
61 \( 1 + (0.362 - 0.931i)T \)
67 \( 1 + (0.516 - 0.856i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.466 + 0.884i)T \)
79 \( 1 + (-0.0285 + 0.999i)T \)
83 \( 1 + (0.736 - 0.676i)T \)
89 \( 1 + (0.998 - 0.0570i)T \)
97 \( 1 + (0.736 + 0.676i)T \)
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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

−17.95250531044397996142299733309, −17.35722800544332409495475044061, −16.47372399319930789848000279412, −16.04955485882147372091699689913, −15.09000279185160286919317522432, −14.55663962285131156367650325212, −13.71399564081538978034952999474, −13.136211085281422088585454723671, −12.4707911579210630867367014995, −11.9133012777033044488678941447, −11.41229724076391062449508992695, −10.63318696520273923037379333, −9.896067482235608491522696921539, −9.16178962169199332948852110820, −7.7075270267800926600548106112, −7.26526545397150465745224842548, −6.72506413756858960300873646816, −5.90507445182878464068311449534, −5.03191041447944843945807077025, −4.75104211029055503981280700856, −3.78349286104527426828889731209, −2.66421848673072217154042080761, −2.014083642711123192525279451085, −1.311314662331623923543631906865, −0.156181553253088776803540513153, 0.861893181793997342492819228030, 2.105921628813221206210711634587, 3.0270331956712107281670581133, 3.76552287708903000002151979251, 4.430255200876414882130182923942, 5.20863235288912032901316345315, 5.808982073615945809537285630639, 6.320200446850827508905460624508, 7.3034201520263025019241315637, 7.852234885736158072616687064030, 9.07081306960563347263865424302, 9.61578071267748908562638886699, 10.82119178562056857609657956991, 11.02206909639862454120597261091, 11.70325200868694581219206093183, 12.5988824524862260476744439958, 12.95014574614659527681988355989, 14.002648274375028108929524747788, 14.52270108724303017926483600296, 15.29570803389715499836145233018, 15.8708375702179304272143832909, 16.38463221602504981710645443071, 17.16309579015380403540889912226, 17.55366235757999593454568506590, 18.52829658315668116164764249336

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