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Properties

Label 1-3503-3503.235-r0-0-0
Degree $1$
Conductor $3503$
Sign $0.915 - 0.403i$
Analytic cond. $16.2678$
Root an. cond. $16.2678$
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  + (0.995 + 0.0896i)2-s + (0.748 − 0.663i)3-s + (0.983 + 0.178i)4-s + (0.399 − 0.916i)5-s + (0.804 − 0.593i)6-s + (−0.842 + 0.538i)7-s + (0.963 + 0.266i)8-s + (0.119 − 0.992i)9-s + (0.480 − 0.877i)10-s + (−0.0299 + 0.999i)11-s + (0.854 − 0.519i)12-s + (0.894 − 0.447i)13-s + (−0.887 + 0.460i)14-s + (−0.309 − 0.951i)15-s + (0.936 + 0.351i)16-s + (0.557 + 0.830i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0896i)2-s + (0.748 − 0.663i)3-s + (0.983 + 0.178i)4-s + (0.399 − 0.916i)5-s + (0.804 − 0.593i)6-s + (−0.842 + 0.538i)7-s + (0.963 + 0.266i)8-s + (0.119 − 0.992i)9-s + (0.480 − 0.877i)10-s + (−0.0299 + 0.999i)11-s + (0.854 − 0.519i)12-s + (0.894 − 0.447i)13-s + (−0.887 + 0.460i)14-s + (−0.309 − 0.951i)15-s + (0.936 + 0.351i)16-s + (0.557 + 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $0.915 - 0.403i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ 0.915 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.039682816 - 1.061374793i\)
\(L(\frac12)\) \(\approx\) \(5.039682816 - 1.061374793i\)
\(L(1)\) \(\approx\) \(2.657080277 - 0.4432642587i\)
\(L(1)\) \(\approx\) \(2.657080277 - 0.4432642587i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.995 + 0.0896i)T \)
3 \( 1 + (0.748 - 0.663i)T \)
5 \( 1 + (0.399 - 0.916i)T \)
7 \( 1 + (-0.842 + 0.538i)T \)
11 \( 1 + (-0.0299 + 0.999i)T \)
13 \( 1 + (0.894 - 0.447i)T \)
17 \( 1 + (0.557 + 0.830i)T \)
19 \( 1 + (0.0970 + 0.995i)T \)
23 \( 1 + (0.674 + 0.738i)T \)
29 \( 1 + (-0.767 + 0.640i)T \)
37 \( 1 + (0.652 - 0.757i)T \)
41 \( 1 + (0.379 - 0.925i)T \)
43 \( 1 + (-0.344 + 0.938i)T \)
47 \( 1 + (0.0224 - 0.999i)T \)
53 \( 1 + (-0.525 - 0.850i)T \)
59 \( 1 + (0.663 + 0.748i)T \)
61 \( 1 + (0.433 - 0.900i)T \)
67 \( 1 + (-0.185 + 0.982i)T \)
71 \( 1 + (0.777 + 0.629i)T \)
73 \( 1 + (0.933 + 0.358i)T \)
79 \( 1 + (-0.685 + 0.727i)T \)
83 \( 1 + (0.575 - 0.817i)T \)
89 \( 1 + (-0.0672 + 0.997i)T \)
97 \( 1 + (-0.963 + 0.266i)T \)
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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

−18.91490499510452129554919605804, −18.52321237526116416215123653474, −17.02691862061563864946376604383, −16.491478277422440142372356344469, −15.859961261031771629426158325894, −15.2810058116630481111878248029, −14.486879808874395023549045608600, −13.90342969985731577218977078336, −13.46025841990752184510769435232, −13.01477829007610978040704182937, −11.662488766297011120105127810373, −10.98498765290626835287684771530, −10.64873772552567254206870882644, −9.70070266601481759433592239844, −9.21010928755155792161497232328, −8.04437607440505548325696943504, −7.25721816463486347883240840836, −6.5301729066417366392325665029, −5.95414510886968995822128475095, −4.99919038903869543222674368372, −4.135180914209722590000389960436, −3.36594166973884846711653707933, −2.998262904576285840295904409818, −2.319877484858673472373914414098, −1.03526782107616057263107012713, 1.18123365838424445478288414392, 1.80391062945299798371572825623, 2.56481201366216926667570824922, 3.63248992765054874136136712147, 3.88231282951941237390374501767, 5.260947427500193057792630305987, 5.733968917442740996049121573110, 6.4426557267272462204999996997, 7.25513899828340008993047360702, 8.03333642682607824211065624265, 8.68169882530617313256085059496, 9.583516592012949705495098106426, 10.13181131567360714214778384853, 11.32956162718369985341354886953, 12.26479898526396464646168897749, 12.73435665868979464889099688188, 12.960271040169332930839990376649, 13.68800507130178398511188768510, 14.567231864579012872096495370209, 15.07437900564956162448862824246, 15.81866774114484891852150466795, 16.43999849472447587739699187792, 17.25288619030008569600869032873, 18.0366043457420740488921347832, 18.87093433961132060686831252606

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