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Properties

Label 2-336-21.5-c1-0-13
Degree $2$
Conductor $336$
Sign $-0.601 + 0.799i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0805 − 1.73i)3-s + (1.90 − 3.29i)5-s + (−2.23 − 1.41i)7-s + (−2.98 + 0.278i)9-s + (−0.309 + 0.178i)11-s + 4.04i·13-s + (−5.84 − 3.02i)15-s + (0.0519 + 0.0900i)17-s + (2.12 + 1.22i)19-s + (−2.26 + 3.98i)21-s + (1.15 + 0.665i)23-s + (−4.72 − 8.17i)25-s + (0.723 + 5.14i)27-s − 4.97i·29-s + (6.83 − 3.94i)31-s + ⋯
L(s)  = 1  + (−0.0465 − 0.998i)3-s + (0.849 − 1.47i)5-s + (−0.844 − 0.535i)7-s + (−0.995 + 0.0929i)9-s + (−0.0933 + 0.0538i)11-s + 1.12i·13-s + (−1.50 − 0.780i)15-s + (0.0126 + 0.0218i)17-s + (0.487 + 0.281i)19-s + (−0.495 + 0.868i)21-s + (0.240 + 0.138i)23-s + (−0.944 − 1.63i)25-s + (0.139 + 0.990i)27-s − 0.923i·29-s + (1.22 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.601 + 0.799i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.601 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558539 - 1.11910i\)
\(L(\frac12)\) \(\approx\) \(0.558539 - 1.11910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.0805 + 1.73i)T \)
7 \( 1 + (2.23 + 1.41i)T \)
good5 \( 1 + (-1.90 + 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.309 - 0.178i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.04iT - 13T^{2} \)
17 \( 1 + (-0.0519 - 0.0900i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.12 - 1.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.15 - 0.665i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + (-6.83 + 3.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.45 + 9.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 + 0.502T + 43T^{2} \)
47 \( 1 + (-5.72 + 9.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.77 - 6.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.20 - 4.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 - 2.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.78iT - 71T^{2} \)
73 \( 1 + (0.203 - 0.117i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.61 + 2.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + (3.41 - 5.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−11.56932457425932665368284003257, −10.05661946988768531537959285305, −9.302791666169699833652537159414, −8.486898865423451503748332145970, −7.34434179137408114756490338306, −6.31536859346745109013955408316, −5.49224863588405686669858451495, −4.16133336571614857754587701062, −2.27304387694422676313683341942, −0.911617935495322735081226830594, 2.82220222494288323897743917578, 3.22229577914149019544579117035, 5.08680823998146928715466956779, 6.03926168134079087813748576181, 6.79204747287978548495099569877, 8.272770821810099872133913596834, 9.485941037271470329576934864757, 10.06286886293297596359720140578, 10.68695355582719948861442276983, 11.59296242356485480276389395262

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