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Properties

Label 2-1078-77.76-c1-0-18
Degree $2$
Conductor $1078$
Sign $0.279 + 0.960i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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 − 2.61i·3-s − 4-s + 2.61i·5-s − 2.61·6-s + i·8-s − 3.82·9-s + 2.61·10-s + (1.41 + 3i)11-s + 2.61i·12-s + 5.54·13-s + 6.82·15-s + 16-s + 3.82i·18-s + 2.29·19-s − 2.61i·20-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.50i·3-s − 0.5·4-s + 1.16i·5-s − 1.06·6-s + 0.353i·8-s − 1.27·9-s + 0.826·10-s + (0.426 + 0.904i)11-s + 0.754i·12-s + 1.53·13-s + 1.76·15-s + 0.250·16-s + 0.902i·18-s + 0.526·19-s − 0.584i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647153090\)
\(L(\frac12)\) \(\approx\) \(1.647153090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (-1.41 - 3i)T \)
good3 \( 1 + 2.61iT - 3T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
13 \( 1 - 5.54T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 + 7.07iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + 6.62iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 12.1iT - 59T^{2} \)
61 \( 1 - 2.29T + 61T^{2} \)
67 \( 1 - 0.343T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 6.49T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 0.951T + 83T^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 - 5.09iT - 97T^{2} \)
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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

−9.803110688639698107345178703030, −8.873146247108139718250751940258, −7.893426160915309926230607382879, −7.19059185303924116813390700165, −6.49076269329826430964191201755, −5.72556548751878163119882139969, −4.12347449994002406033147335457, −3.09748716557972960166668854051, −2.11601236355239761053763034663, −1.14550722129825163928015608082, 0.979977465996590940251413101929, 3.31313636448475616546623741918, 4.13915982788389670591475330769, 4.77294229357464434856449464226, 5.76308774794183395247499392963, 6.27459026661676471766735934076, 8.000293008647862215810827579889, 8.436824972331246426888009619958, 9.367360604235272030825576491433, 9.559841440186483509583457019214

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