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Properties

Label 2-336-336.173-c1-0-25
Degree $2$
Conductor $336$
Sign $0.992 + 0.121i$
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 + (1.44 + 0.961i)3-s + (−1.96 − 0.382i)4-s + (0.495 + 1.84i)5-s + (1.54 − 1.89i)6-s + (2.64 − 0.164i)7-s + (−0.805 + 2.71i)8-s + (1.15 + 2.76i)9-s + (2.66 − 0.445i)10-s + (−0.781 + 2.91i)11-s + (−2.46 − 2.43i)12-s + (−1.32 + 1.32i)13-s + (0.126 − 3.73i)14-s + (−1.06 + 3.13i)15-s + (3.70 + 1.50i)16-s + (−1.97 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.0960 − 0.995i)2-s + (0.831 + 0.554i)3-s + (−0.981 − 0.191i)4-s + (0.221 + 0.826i)5-s + (0.632 − 0.774i)6-s + (0.998 − 0.0622i)7-s + (−0.284 + 0.958i)8-s + (0.384 + 0.923i)9-s + (0.843 − 0.140i)10-s + (−0.235 + 0.879i)11-s + (−0.710 − 0.703i)12-s + (−0.368 + 0.368i)13-s + (0.0339 − 0.999i)14-s + (−0.274 + 0.810i)15-s + (0.926 + 0.375i)16-s + (−0.480 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.992 + 0.121i$
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: \((2,\ 336,\ (\ :1/2),\ 0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77638 - 0.107949i\)
\(L(\frac12)\) \(\approx\) \(1.77638 - 0.107949i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 - 0.961i)T \)
7 \( 1 + (-2.64 + 0.164i)T \)
good5 \( 1 + (-0.495 - 1.84i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.781 - 2.91i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.32 - 1.32i)T - 13iT^{2} \)
17 \( 1 + (1.97 + 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.595 + 2.22i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.20 + 5.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.37 + 3.37i)T - 29iT^{2} \)
31 \( 1 + (0.517 - 0.298i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.37 + 8.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 1.57iT - 41T^{2} \)
43 \( 1 + (9.17 - 9.17i)T - 43iT^{2} \)
47 \( 1 + (-0.138 + 0.240i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.638 + 2.38i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (11.6 + 3.11i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.203 + 0.759i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.82 - 6.82i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (6.33 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.42 - 2.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.60 - 9.60i)T + 83iT^{2} \)
89 \( 1 + (10.9 + 6.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.81iT - 97T^{2} \)
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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

−11.23131616149622146898765820094, −10.66911189304388623014552789530, −9.821641195089347652709518295195, −9.003255114878074383575481030720, −8.021963459055190285468148810995, −6.90722508531370226775654924942, −4.94213693505462617692324871916, −4.42579587466364644174544939945, −2.86738050297379831632972698247, −2.09769731881649901995385428504, 1.38310998686283399639452181174, 3.39729838865446154724671128996, 4.77303144145003222098659457736, 5.68214795064783524256321495461, 6.91418119976312365424667110957, 7.994749962830085409119079702456, 8.496412453631134343119156316471, 9.165127663922360850112745416467, 10.43926926425700651385808541730, 11.91990572963175153578645924995

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