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Properties

Label 2-3332-476.135-c0-0-10
Degree $2$
Conductor $3332$
Sign $-0.266 + 0.963i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s + 2·13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)26-s + (0.499 + 0.866i)32-s + 0.999·34-s − 0.999·36-s − 0.999·50-s + (−0.999 − 1.73i)52-s + (−1 − 1.73i)53-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (0.5 − 0.866i)9-s + 2·13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.5 − 0.866i)25-s + (1 − 1.73i)26-s + (0.499 + 0.866i)32-s + 0.999·34-s − 0.999·36-s − 0.999·50-s + (−0.999 − 1.73i)52-s + (−1 − 1.73i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.266 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.698712486\)
\(L(\frac12)\) \(\approx\) \(1.698712486\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
17 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 2T + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.612420346822928727030383580695, −8.176187904380130979774471702948, −6.78421467486257145275805207514, −6.17516738020270335602904850429, −5.62264660835330541231651592472, −4.43172519368160728415277307891, −3.77334980129548481202183683420, −3.24897868620786090643291514224, −1.87312827483466508110981069627, −1.02368798150210056960391418729, 1.43444400753966688670149187750, 2.87969008200034055028423249611, 3.72845319858894840005191816088, 4.47887448849527279390066982580, 5.36332984954896989788059762674, 5.95236142671556961796155853963, 6.76629422866384744025966600858, 7.56244060526558704183467388904, 8.052473101106199130887091473846, 8.870431060253199522193022083744

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