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Properties

Label 1-1157-1157.1004-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.334 - 0.942i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.235 − 0.971i)2-s + (0.888 − 0.458i)3-s + (−0.888 − 0.458i)4-s + (−0.654 + 0.755i)5-s + (−0.235 − 0.971i)6-s + (−0.981 − 0.189i)7-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)9-s + (0.580 + 0.814i)10-s + (0.981 − 0.189i)11-s − 12-s + (−0.415 + 0.909i)14-s + (−0.235 + 0.971i)15-s + (0.580 + 0.814i)16-s + (0.235 + 0.971i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (0.888 − 0.458i)3-s + (−0.888 − 0.458i)4-s + (−0.654 + 0.755i)5-s + (−0.235 − 0.971i)6-s + (−0.981 − 0.189i)7-s + (−0.654 + 0.755i)8-s + (0.580 − 0.814i)9-s + (0.580 + 0.814i)10-s + (0.981 − 0.189i)11-s − 12-s + (−0.415 + 0.909i)14-s + (−0.235 + 0.971i)15-s + (0.580 + 0.814i)16-s + (0.235 + 0.971i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.334 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413483370 - 0.9986206814i\)
\(L(\frac12)\) \(\approx\) \(1.413483370 - 0.9986206814i\)
\(L(1)\) \(\approx\) \(1.098964662 - 0.6205699775i\)
\(L(1)\) \(\approx\) \(1.098964662 - 0.6205699775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.235 - 0.971i)T \)
3 \( 1 + (0.888 - 0.458i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.981 - 0.189i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
17 \( 1 + (0.235 + 0.971i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.327 + 0.945i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.0475 - 0.998i)T \)
43 \( 1 + (0.327 - 0.945i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.981 + 0.189i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.47648444795742143979543490264, −20.69809524478609314735619908124, −19.76334424214647171811873909691, −19.262276850989738877951144177116, −18.46904930545535947936896830324, −17.11577486752068103037930044431, −16.605031783771674558293583706645, −15.929439970699612922907599376795, −15.224322909010645431760257130090, −14.74430331273327454387184952756, −13.58574027386632553691238043122, −13.1469055363898943784301768032, −12.31465847184930085599385461459, −11.3420662237699718814312054164, −9.76214671826023906367156240304, −9.31196510667905689427321099991, −8.73387940234173327857506117538, −7.819434191685469540308731923770, −7.06172308676720034955497990287, −6.17988389468388956637035738334, −4.916229133796978144017182134458, −4.35644901173455991297876553781, −3.5269390632596977494498499792, −2.6680745456079562767414775466, −0.814236898208601047085414697820, 0.91734321206320820601206102550, 2.02276651002955358760446251906, 3.03101528708013987404095633504, 3.68223565751538359083559182610, 4.098985443844288024554780717543, 5.82251648420237854358661251347, 6.69346792494085720557104272891, 7.463544262883303285475034247542, 8.72634740918462358570567572110, 9.052545305781614606332803349645, 10.32342741240145908374534870356, 10.65668831843545577093261326989, 12.01724581729258083617320488042, 12.3948046464675106962572425749, 13.22653564458670627998012911058, 14.114323191587752945971380190291, 14.664230785704990067005907921292, 15.306666476646891497922068504605, 16.496116774290166844487177559321, 17.54599129131294732583031139529, 18.566273796006956610301769528505, 19.08293425395914751528290746237, 19.586371418968164223542174976444, 20.02462391798440323190618530076, 21.07918276160656124930807057100

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