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Properties

Label 2-336-336.173-c1-0-26
Degree $2$
Conductor $336$
Sign $0.927 - 0.373i$
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  + (−1.29 − 0.576i)2-s + (1.68 + 0.380i)3-s + (1.33 + 1.48i)4-s + (0.930 + 3.47i)5-s + (−1.96 − 1.46i)6-s + (0.200 − 2.63i)7-s + (−0.865 − 2.69i)8-s + (2.70 + 1.28i)9-s + (0.800 − 5.01i)10-s + (1.13 − 4.23i)11-s + (1.68 + 3.02i)12-s + (−1.18 + 1.18i)13-s + (−1.77 + 3.29i)14-s + (0.249 + 6.22i)15-s + (−0.435 + 3.97i)16-s + (2.65 + 4.60i)17-s + ⋯
L(s)  = 1  + (−0.913 − 0.407i)2-s + (0.975 + 0.219i)3-s + (0.667 + 0.744i)4-s + (0.416 + 1.55i)5-s + (−0.801 − 0.598i)6-s + (0.0756 − 0.997i)7-s + (−0.305 − 0.952i)8-s + (0.903 + 0.429i)9-s + (0.253 − 1.58i)10-s + (0.342 − 1.27i)11-s + (0.487 + 0.873i)12-s + (−0.329 + 0.329i)13-s + (−0.475 + 0.879i)14-s + (0.0644 + 1.60i)15-s + (−0.108 + 0.994i)16-s + (0.645 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31405 + 0.254725i\)
\(L(\frac12)\) \(\approx\) \(1.31405 + 0.254725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.576i)T \)
3 \( 1 + (-1.68 - 0.380i)T \)
7 \( 1 + (-0.200 + 2.63i)T \)
good5 \( 1 + (-0.930 - 3.47i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.13 + 4.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.18 - 1.18i)T - 13iT^{2} \)
17 \( 1 + (-2.65 - 4.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.533 - 1.99i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.573 + 0.993i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.354 + 0.354i)T - 29iT^{2} \)
31 \( 1 + (7.76 - 4.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.05 + 3.94i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.75iT - 41T^{2} \)
43 \( 1 + (-5.87 + 5.87i)T - 43iT^{2} \)
47 \( 1 + (4.63 - 8.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.58 + 5.91i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.27 + 1.94i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.845 - 3.15i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.82 + 14.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.726T + 71T^{2} \)
73 \( 1 + (6.35 + 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.16 + 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.06 + 6.06i)T + 83iT^{2} \)
89 \( 1 + (-10.0 - 5.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.30iT - 97T^{2} \)
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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

−10.89648672894281303535406506435, −10.73356879169542314867473468601, −9.875620128496882172306533140967, −8.925962141332738244952644537250, −7.84639734553813490322588221423, −7.16911639123752946785537981555, −6.20017532075452276283887771792, −3.74536741629959780775823922456, −3.22674199374955554942050073922, −1.80233880788995590050257396343, 1.39149205324214075026607521111, 2.52286146010768676333077227873, 4.72211895800898379575585313255, 5.59241920468573658712859750825, 7.06414595628097677354438742350, 7.931390583734135840207424011211, 8.826537325697990258173043743041, 9.469972421361485353336347211524, 9.814722652937981965862557831896, 11.66330228330971878467365435765

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