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Properties

Label 2-336-112.27-c1-0-25
Degree $2$
Conductor $336$
Sign $-0.903 + 0.429i$
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  + (−0.135 − 1.40i)2-s + (−0.707 + 0.707i)3-s + (−1.96 + 0.381i)4-s + (2.46 − 2.46i)5-s + (1.09 + 0.899i)6-s + (−2.50 − 0.865i)7-s + (0.803 + 2.71i)8-s − 1.00i·9-s + (−3.80 − 3.13i)10-s + (0.244 − 0.244i)11-s + (1.11 − 1.65i)12-s + (−2.35 − 2.35i)13-s + (−0.878 + 3.63i)14-s + 3.48i·15-s + (3.70 − 1.49i)16-s − 5.19i·17-s + ⋯
L(s)  = 1  + (−0.0959 − 0.995i)2-s + (−0.408 + 0.408i)3-s + (−0.981 + 0.190i)4-s + (1.10 − 1.10i)5-s + (0.445 + 0.367i)6-s + (−0.945 − 0.326i)7-s + (0.284 + 0.958i)8-s − 0.333i·9-s + (−1.20 − 0.992i)10-s + (0.0737 − 0.0737i)11-s + (0.322 − 0.478i)12-s + (−0.652 − 0.652i)13-s + (−0.234 + 0.972i)14-s + 0.900i·15-s + (0.927 − 0.374i)16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.903 + 0.429i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.903 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191058 - 0.846797i\)
\(L(\frac12)\) \(\approx\) \(0.191058 - 0.846797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.135 + 1.40i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (2.50 + 0.865i)T \)
good5 \( 1 + (-2.46 + 2.46i)T - 5iT^{2} \)
11 \( 1 + (-0.244 + 0.244i)T - 11iT^{2} \)
13 \( 1 + (2.35 + 2.35i)T + 13iT^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 + (-1.43 + 1.43i)T - 19iT^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + (5.09 - 5.09i)T - 29iT^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 + (2.40 + 2.40i)T + 37iT^{2} \)
41 \( 1 - 9.55T + 41T^{2} \)
43 \( 1 + (-6.80 + 6.80i)T - 43iT^{2} \)
47 \( 1 - 7.20T + 47T^{2} \)
53 \( 1 + (-5.49 - 5.49i)T + 53iT^{2} \)
59 \( 1 + (3.45 + 3.45i)T + 59iT^{2} \)
61 \( 1 + (1.27 + 1.27i)T + 61iT^{2} \)
67 \( 1 + (-3.83 - 3.83i)T + 67iT^{2} \)
71 \( 1 - 16.0T + 71T^{2} \)
73 \( 1 - 6.68T + 73T^{2} \)
79 \( 1 + 4.09iT - 79T^{2} \)
83 \( 1 + (9.70 - 9.70i)T - 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−11.01955450706232818932686038815, −10.10351225041648414285957840996, −9.465747349198552886523344725640, −8.997584763153720833226402544504, −7.43684107943634104931580630043, −5.76914149313016410860423943866, −5.14213464095302458909816424992, −3.91606710949544645205411826316, −2.45309652046572964929662830970, −0.65657874178993644358330074197, 2.20029681030021907687912845638, 3.93212413857080102386834206068, 5.72539097990243420249606832815, 6.12536693787273824431969197186, 6.91306786858734939999797976357, 7.88222142808578719027364240647, 9.351444220787326760963767325690, 9.872580006476932354205472296170, 10.74787411316384496041775212884, 12.18965035699340105121179839867

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