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Properties

Label 1-4235-4235.107-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.523 - 0.851i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.997 + 0.0665i)2-s + (−0.994 + 0.104i)3-s + (0.991 − 0.132i)4-s + (0.985 − 0.170i)6-s + (−0.980 + 0.198i)8-s + (0.978 − 0.207i)9-s + (−0.971 + 0.235i)12-s + (−0.825 − 0.564i)13-s + (0.964 − 0.263i)16-s + (−0.603 − 0.797i)17-s + (−0.962 + 0.272i)18-s + (−0.290 − 0.956i)19-s + (−0.998 − 0.0475i)23-s + (0.953 − 0.299i)24-s + (0.861 + 0.508i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.997 + 0.0665i)2-s + (−0.994 + 0.104i)3-s + (0.991 − 0.132i)4-s + (0.985 − 0.170i)6-s + (−0.980 + 0.198i)8-s + (0.978 − 0.207i)9-s + (−0.971 + 0.235i)12-s + (−0.825 − 0.564i)13-s + (0.964 − 0.263i)16-s + (−0.603 − 0.797i)17-s + (−0.962 + 0.272i)18-s + (−0.290 − 0.956i)19-s + (−0.998 − 0.0475i)23-s + (0.953 − 0.299i)24-s + (0.861 + 0.508i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.523 - 0.851i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.523 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1730037559 - 0.3095128169i\)
\(L(\frac12)\) \(\approx\) \(0.1730037559 - 0.3095128169i\)
\(L(1)\) \(\approx\) \(0.4540245527 - 0.04370603689i\)
\(L(1)\) \(\approx\) \(0.4540245527 - 0.04370603689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.997 + 0.0665i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.825 - 0.564i)T \)
17 \( 1 + (-0.603 - 0.797i)T \)
19 \( 1 + (-0.290 - 0.956i)T \)
23 \( 1 + (-0.998 - 0.0475i)T \)
29 \( 1 + (0.0855 + 0.996i)T \)
31 \( 1 + (-0.851 - 0.524i)T \)
37 \( 1 + (0.983 - 0.179i)T \)
41 \( 1 + (0.466 + 0.884i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.962 + 0.272i)T \)
53 \( 1 + (0.263 - 0.964i)T \)
59 \( 1 + (0.532 + 0.846i)T \)
61 \( 1 + (0.830 + 0.556i)T \)
67 \( 1 + (0.618 - 0.786i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.151 + 0.988i)T \)
79 \( 1 + (-0.595 - 0.803i)T \)
83 \( 1 + (-0.633 + 0.774i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (0.676 - 0.736i)T \)
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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

−18.61030505521177973228334652658, −17.79729402740442377915024800120, −17.2459523429995925967576811598, −16.83603203258833569179832411977, −16.0747789722976252981383961116, −15.55441599356838425188700792813, −14.72355378657866241208795853359, −13.90761796167557273885023367522, −12.609695592875748591922447848334, −12.48068198840537473801807492663, −11.57207678015492118008427880200, −11.080895115645234373677450415998, −10.24306315865531267353860995166, −9.88501479253081683728258073662, −9.007425017398859897353281673731, −8.16488985859142300914422008690, −7.443627289015104718768240905563, −6.847576255568426368263947712531, −6.01275536644314647419913624250, −5.61662269053407700043465979842, −4.34362773896085490684156837558, −3.80343218069276817889796527584, −2.304498536863036392426632225784, −1.92452840462737220556085284994, −0.84458975355429385516110549083, 0.22897842600427162641001247312, 0.99070691374668711607082647096, 2.14660481098489283216229034478, 2.77587752291616103656881216422, 4.01979836725368623501024703719, 4.88702462835709251520593020474, 5.63206564997141098911021375424, 6.31259363444137151071911631409, 7.158150765108807888123150362072, 7.49311044688373269694103695526, 8.51009776902581845809819439756, 9.39268753690791861389361441294, 9.812505630605404809576006059101, 10.63766259985713596431844204027, 11.19937244549730372712331960912, 11.719320684217141143157814465186, 12.56882396194122135337217395041, 13.059209665946534559634060943323, 14.30020892009743107989582736486, 15.04489514999477562226792968664, 15.70920638989912633709938114166, 16.280589117394151487368382985875, 16.839756392167426602308716906235, 17.621765830298605713840225308497, 18.00620907677254125192871632272

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