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Properties

Label 2-338-13.12-c1-0-5
Degree $2$
Conductor $338$
Sign $0.0304 - 0.999i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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  + i·2-s + 2.35·3-s − 4-s + 0.890i·5-s + 2.35i·6-s + 4.49i·7-s − i·8-s + 2.55·9-s − 0.890·10-s − 2.69i·11-s − 2.35·12-s − 4.49·14-s + 2.09i·15-s + 16-s − 3.58·17-s + 2.55i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.36·3-s − 0.5·4-s + 0.398i·5-s + 0.962i·6-s + 1.69i·7-s − 0.353i·8-s + 0.851·9-s − 0.281·10-s − 0.811i·11-s − 0.680·12-s − 1.20·14-s + 0.541i·15-s + 0.250·16-s − 0.868·17-s + 0.602i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.0304 - 0.999i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.0304 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34219 + 1.30188i\)
\(L(\frac12)\) \(\approx\) \(1.34219 + 1.30188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
13 \( 1 \)
good3 \( 1 - 2.35T + 3T^{2} \)
5 \( 1 - 0.890iT - 5T^{2} \)
7 \( 1 - 4.49iT - 7T^{2} \)
11 \( 1 + 2.69iT - 11T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + 2.93iT - 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 1.50iT - 37T^{2} \)
41 \( 1 + 3.65iT - 41T^{2} \)
43 \( 1 + 0.170T + 43T^{2} \)
47 \( 1 + 5.20iT - 47T^{2} \)
53 \( 1 - 6.09T + 53T^{2} \)
59 \( 1 + 3.07iT - 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 - 2.93iT - 83T^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−11.85640433446130502167651413798, −10.77504573192763220100176117260, −9.270824225578546871512975247830, −8.882270577871141719572348289069, −8.282874339280578244700172004824, −7.06726461701306438080338256548, −6.05068560950295705100068242991, −4.89404505612156742491595940295, −3.24439438021732674384514175994, −2.44457143953595459335701192797, 1.36532783804410111046656701360, 2.86455585472134964059290587836, 3.98611647356628423337514117727, 4.75134476885713122634305475743, 6.86236379213455161399702346428, 7.70569317492863907635810165615, 8.662247785862802145917985596001, 9.470068944905294044316091223976, 10.33242152266622065880616166804, 11.09321996409962345467663912464

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