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Properties

Label 2-3332-3332.2515-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.481 + 0.876i$
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  + (−0.365 − 0.930i)2-s + (−0.149 − 1.98i)3-s + (−0.733 + 0.680i)4-s + (−1.79 + 0.865i)6-s + (−0.433 + 0.900i)7-s + (0.900 + 0.433i)8-s + (−2.94 + 0.443i)9-s + (1.84 + 0.277i)11-s + (1.46 + 1.35i)12-s + (−0.455 + 0.571i)13-s + (0.997 + 0.0747i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (1.48 + 2.57i)18-s + (1.85 + 0.728i)21-s + (−0.414 − 1.81i)22-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.149 − 1.98i)3-s + (−0.733 + 0.680i)4-s + (−1.79 + 0.865i)6-s + (−0.433 + 0.900i)7-s + (0.900 + 0.433i)8-s + (−2.94 + 0.443i)9-s + (1.84 + 0.277i)11-s + (1.46 + 1.35i)12-s + (−0.455 + 0.571i)13-s + (0.997 + 0.0747i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (1.48 + 2.57i)18-s + (1.85 + 0.728i)21-s + (−0.414 − 1.81i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9249834998\)
\(L(\frac12)\) \(\approx\) \(0.9249834998\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (0.433 - 0.900i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
good3 \( 1 + (0.149 + 1.98i)T + (-0.988 + 0.149i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-1.84 - 0.277i)T + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.455 - 0.571i)T + (-0.222 - 0.974i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.49 - 0.460i)T + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0747 - 0.997i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (-0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.250 - 1.09i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (-0.149 + 0.258i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.147 + 0.0222i)T + (0.955 - 0.294i)T^{2} \)
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

−8.754544724297870351345490955772, −7.84738728630060470723852843624, −7.02651647684643329954788268327, −6.60972766400120352272251827049, −5.70803714724611779299667061271, −4.74191036026273193172260542667, −3.31606604418325077881974066859, −2.69026325392106622771774940414, −1.71231673097765192992542225312, −1.03774895530056787745488036532, 0.884543582747539532695854431845, 3.27379726588034037706103030928, 3.72050301161578072520764466380, 4.57284103610135599657559904490, 5.14783706370095813021331311642, 6.05739409530682606607523932561, 6.63020090545962215425442197434, 7.63924988433357425389075627848, 8.556485924309393959996671871375, 9.191474060021538723185689281640

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