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Properties

Label 2-368-16.13-c1-0-15
Degree $2$
Conductor $368$
Sign $0.835 - 0.549i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (−1.27 + 0.602i)2-s + (0.203 + 0.203i)3-s + (1.27 − 1.54i)4-s + (1.88 − 1.88i)5-s + (−0.383 − 0.137i)6-s + 4.71i·7-s + (−0.702 + 2.73i)8-s − 2.91i·9-s + (−1.27 + 3.54i)10-s + (−0.944 + 0.944i)11-s + (0.573 − 0.0543i)12-s + (1.72 + 1.72i)13-s + (−2.83 − 6.03i)14-s + 0.766·15-s + (−0.751 − 3.92i)16-s + 7.35·17-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)2-s + (0.117 + 0.117i)3-s + (0.637 − 0.770i)4-s + (0.841 − 0.841i)5-s + (−0.156 − 0.0562i)6-s + 1.78i·7-s + (−0.248 + 0.968i)8-s − 0.972i·9-s + (−0.402 + 1.11i)10-s + (−0.284 + 0.284i)11-s + (0.165 − 0.0156i)12-s + (0.478 + 0.478i)13-s + (−0.758 − 1.61i)14-s + 0.197·15-s + (−0.187 − 0.982i)16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06407 + 0.318567i\)
\(L(\frac12)\) \(\approx\) \(1.06407 + 0.318567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.602i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-0.203 - 0.203i)T + 3iT^{2} \)
5 \( 1 + (-1.88 + 1.88i)T - 5iT^{2} \)
7 \( 1 - 4.71iT - 7T^{2} \)
11 \( 1 + (0.944 - 0.944i)T - 11iT^{2} \)
13 \( 1 + (-1.72 - 1.72i)T + 13iT^{2} \)
17 \( 1 - 7.35T + 17T^{2} \)
19 \( 1 + (1.62 + 1.62i)T + 19iT^{2} \)
29 \( 1 + (1.16 + 1.16i)T + 29iT^{2} \)
31 \( 1 - 9.98T + 31T^{2} \)
37 \( 1 + (3.26 - 3.26i)T - 37iT^{2} \)
41 \( 1 - 5.41iT - 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 + (-7.79 + 7.79i)T - 53iT^{2} \)
59 \( 1 + (-6.83 + 6.83i)T - 59iT^{2} \)
61 \( 1 + (-1.70 - 1.70i)T + 61iT^{2} \)
67 \( 1 + (9.77 + 9.77i)T + 67iT^{2} \)
71 \( 1 - 0.932iT - 71T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + (4.25 + 4.25i)T + 83iT^{2} \)
89 \( 1 + 0.581iT - 89T^{2} \)
97 \( 1 + 9.37T + 97T^{2} \)
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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

−11.65988317452632353201690123643, −9.990792647802070196752771965341, −9.583818140395529701128172353729, −8.733601900967680798051110093630, −8.215672520435895241346336199859, −6.55130571583917741727416437485, −5.83230248902318969004168828300, −5.03670642108971078905327971635, −2.81082021004663622344326400844, −1.44394112667617593768511346955, 1.24750137140108481252106384860, 2.76229636406048027532304626742, 3.86457519927869012694671018940, 5.70205442743092412033065746288, 6.94959718508441058541689989599, 7.61031468710999675672671492461, 8.415476074694278076018699298310, 10.01985772414975325018763903018, 10.36692158061112043235457888494, 10.74321091354257789681941862757

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